Chapter I - Many Christians Are Confused About What our Relationship to the Old Testament And Its Laws Should Be.
Chapter II - What The New Testament Says About Old Testament Law
Chapter III - Jesus and The Spirit or Intention Behind The Law
ON THIS PAGE
Note: Unless otherwise stated all Biblical quotes are from the NASB
The Book of Hebrews and The Law
Hebrews can be divided into sections, each of which details Jesus' superiority over various ancients, the temple sacrifices, and the Old Testament covenant itself.
Corroborating New Testament Verses
The Book of Hebrews and The Law
Many centuries ago, the prophet Jeremiah told us that the Father was going to make a new covenant with the house of Israel and Judah. One in which His laws would be written on their hearts, not taught them. Here is the prophecy he made
"Behold, days are coming," declares the Lord, "when I will make a new covenant with the house of Israel and with the house of Judah, not like the covenant which I made with their fathers in the day I took them by the hand to bring them out of the land of Egypt, My covenant which they broke, although I was a husband to them," declares the Lord. "But this is the covenant which I will make with the house of Israel after those days," declares the Lord, "I will put My law within them and on their heart I will write it; and I will be their God, and they shall be My people. "They will not teach again, each man his neighbor and each man his brother, saying, 'Know the Lord,' for they will all know Me, from the least of them to the greatest of them," declares the Lord, "for I will forgive their iniquity, and their sin I will remember no more." (Jeremiah 31:31-34 NASB)
Fast forward a few centuries to the time after Christ's death and ascension when someone, whose identity is yet concealed from us, sat down and penned one of the most fascinating and detailed accounts of the superiority of Christ and the New Covenant. In fact, the entire purpose of the Book of Hebrews was to help Jewish Christians understand that the old covenant was superseded by something far greater - this clearly demonstrated by the opening words of the book.
God, after He spoke long ago to the fathers in the prophets in many portions and in many ways, in these last days has spoken to us in His Son, whom He appointed heir of all things, through whom also He made the world. And He is the radiance of His glory and the exact representation of His nature, and upholds all things by the word of His power. When He had made purification of sins, He sat down at the right hand of the Majesty on high, (Hebrews 1:1-3 NASB)
Hebrews can be divided into sections, each of which details Jesus' superiority over various ancients, the temple sacrifices, and the Old Testament covenant itself. Each section also usually includes a warning.
Note: The points below have been condensed from an excellent article entitled Hebrews: Introduction, Argument, and Outline by Daniel B. Wallace - Points 4 and 5 being direct quotations. As he wrote, the author of Hebrews contrasts the old covenant with the new (8:7-13). The inadequacy of the old covenant is demonstrated by scripture (8:7-9), as is the adequacy of the new covenant. (8:10-13).  The entire article is well worth reading HERE
1) Christ is Superior to the prophets (1:1-4). The prophets were servants or spokespersons of the Lord. Christ is the Son of God.
2) Christ is superior to the angels (1:5–2:18) inasmuch as He has inherited a more excellent name than the angels.
3) Christ is superior to Moses (3:1–4:13). The faithful Moses was a servant of the Father. Christ was faithful as a Son over His house"
4) "Christ is superior to Aaron (4:14–7:28). The priesthood of Aaron is first mentioned (5:1-5), followed by scriptural proof (based especially on Psalm 110) for the priesthood of Christ (5:6-10) after the order of Melchizedek (5:6, 10) - proof which is necessary since Jesus Christ was not from the tribe of Levi." 
5) Christ's ministry is superior to the old covenant ministry (8:1–10:18).
It is the last point (5) that particularly concerns us in the context of this article. In chapter 8, the author of Hebrews pointed to Christ who, as the high priest, has taken His seat at the right hand of the throne of the Majesty in the heavens. A few verses later he quoted Jeremiah 31:31-34 sandwiched between two very telling remarks -
But now He has obtained a more excellent ministry, by as much as He is also the mediator of a better covenant, which has been enacted on better promises. For if that first covenant had been faultless, there would have been no occasion sought for a second.
"Behold, days are coming," declares the LORD, "when I will make a new covenant with the house of Israel and with the house of Judah, not like the covenant which I made with their fathers in the day I took them by the hand to bring them out of the land of Egypt, My covenant which they broke, although I was a husband to them," declares the LORD. "But this is the covenant which I will make with the house of Israel after those days," declares the LORD, "I will put My law within them and on their heart I will write it; and I will be their God, and they shall be My people. "They will not teach again, each man his neighbor and each man his brother, saying, 'Know the LORD,' for they will all know Me, from the least of them to the greatest of them," declares the LORD, "for I will forgive their iniquity, and their sin I will remember no more." (This section is Jeremiah’s Prophecy in Jeremiah 31:31-34)
When He said, "A new covenant," He has made the first obsolete. But whatever is becoming obsolete and growing old is ready to disappear. (Hebrews 8:6-13 NASB)
The essence of Jeremiah's words were once again reiterated in 10:15-20
Note: The words "whatever is becoming obsolete and growing old is ready to disappear" in Verse 13 are extremely telling in view of the fact that the temple in Jerusalem (indispensable to the proper keeping of many, if not most, of the Old Testament rituals) was destroyed very shortly after the book was written. How do we know? See Footnote
Hebrews also states
For, on the one hand, there is a setting aside of a former commandment because of its weakness and uselessness (for the Law made nothing perfect), and on the other hand there is a bringing in of a better hope, through which we draw near to God. (Hebrews 7:18-19 NASB)
in whole burnt offerings and sacrifices for sin you have taken no pleasure. "then I said, 'behold, I have come (in the scroll of the book it is written of Me) to do Your will, o God.'" after saying above, "sacrifices and offerings and whole burnt offerings and sacrifices for sin You have not desired, nor have You taken pleasure in them" (which are offered according to the law), then He said, "behold, I have come to do Your will." He takes away the first in order to establish the second. (Hebrews 10:6-9 NASB)
In fact, the author of Hebrews summed it up perfectly when he said
(22) so much the more also Jesus has become the guarantee of a better covenant. (23) The former priests, on the one hand, existed in greater numbers because they were prevented by death from continuing, (24) but Jesus, on the other hand, because He continues forever, holds His priesthood permanently. (25) Therefore He is able also to save forever those who draw near to God through Him, since He always lives to make intercession for them. (26) For it was fitting for us to have such a high priest, holy, innocent, undefiled, separated from sinners and exalted above the heavens; (27) who does not need daily, like those high priests, to offer up sacrifices, first for His own sins and then for the sins of the people, because this He did once for all when He offered up Himself. (28) For the Law appoints men as high priests who are weak, but the word of the oath, which came after the Law, appoints a Son, made perfect forever. (Hebrews 7:22-28 NASB)
So we need to understand that the death and resurrection of Christ ushered in the New Covenant and, in doing so, rendered the Old Covenant obsolete.
Corroborating New Testament Verses
It seems readily apparent that Jewish rites were observed for some period of time by those Christians who had converted from Judaism. However, while Paul expressly declared that he himself kept the law, saying "Neither against the law of the Jews, nor against the temple, nor against Caesar, have I sinned at all" (Acts 25:8) he was, at the same time, very emphatic that we were no long under the law.
Galatians: In fact, the book of Galatians is almost exclusively devoted to the fact that salvation is acquired through grace not law. Paul emphatically told the Galatians that since they were no longer under bondage to the dictates of the Old Testament Law, they needed to stop listening to the false teachers who would do nothing but once again fatally entangle them in rules and regulations that could not save. As he said
You foolish Galatians, who has bewitched you, before whose eyes Jesus Christ was publicly portrayed as crucified? This is the only thing I want to find out from you: did you receive the Spirit by the works of the Law, or by hearing with faith? Are you so foolish? Having begun by the Spirit, are you now being perfected by the flesh? Did you suffer so many things in vain--if indeed it was in vain? (Galatians 3:1-4 NASB)
Additional statements include
for if righteousness come by the law, then Christ is dead in vain. [Galatians 2:21]
For as many as are of the works of the Law are under a curse; for it is written, "cursed is everyone who does not abide by all things written in the book of the law, to perform them." now that no one is justified by the law before God is evident; for, "the righteous man shall live by faith." however, the Law is not of faith; on the contrary, "he who practices them shall live by them." Christ redeemed us from the curse of the law, having become a curse for us--for it is written, "cursed is everyone who hangs on a tree"-- (Galatians 3:10-13 NASB)
But before faith came, we were kept in custody under the law, being shut up to the faith which was later to be revealed. Therefore the Law has become our tutor to lead us to Christ, so that we may be justified by faith. But now that faith has come, we are no longer under a tutor. (Galatians 3:23-25 NASB)
But now that you have come to know God, or rather to be known by God, how is it that you turn back again to the weak and worthless elemental things, to which you desire to be enslaved all over again? You observe days and months and seasons and years. I fear for you, that perhaps I have labored over you in vain. (Galatians 4:9-11 NASB)
You have been severed from Christ, you who are seeking to be justified by law; you have fallen from grace. (Galatians 5:4 NASB)
But if you are led by the Spirit, you are not under the Law. (Galatians 5:18 NASB)
In fact, Paul went as far as to reprimand Peter for compelling the Gentiles to live as Jews - After the vision recorded in Acts 10, Peter was convinced that God had pulled down the middle wall of partition that had so long separated the Jews and Gentiles. He ate with the latter, evidently disregarding the dietary laws of the Jews. However when certain Jews came from Jerusalem he withdrew from the Gentiles and appeared to be enforcing the distinction between the two groups. However, this resulted in him being chided by Paul, not for eating with the Gentiles, but for compelling the Gentiles to live as Jews, although he knew that a man is not justified by the works of the law but through faith in Jesus Christ [V.16]
But when Cephas came to Antioch, I opposed him to his face, because he stood condemned. For prior to the coming of certain men from James, he used to eat with the Gentiles; but when they came, he began to withdraw and hold himself aloof, fearing the party of the circumcision. The rest of the Jews joined him in hypocrisy, with the result that even Barnabas was carried away by their hypocrisy. But when I saw that they were not straightforward about the truth of the gospel, I said to Cephas in the presence of all, "If you, being a Jew, live like the Gentiles and not like the Jews, how is it that you compel the Gentiles to live like Jews? (Galatians 2:11-14 NASB)
Ephesians: The Old Testament covenant, of which the Law was a tangible sign, was made between the Jews and God. The covenant signified the special relationship the nation of Israel had with the Father - a status that only a handful of gentiles ever achieved.
In chapter 2, verses 13 and 14, Paul told the largely Gentile church that they who were formerly far off - separated from the relationship with the Father enjoyed by the Israelites had been brought near by the blood of Christ. He went on to say that Christ Himself is our peace, who made both groups into one and broke down the barrier of the dividing wall. Paul then emphasized the fact that the Law that separated Jew from Gentile, was no longer in effect.
by abolishing in His flesh the enmity, which is the Law of commandments contained in ordinances, so that in Himself He might make the two into one new man, thus establishing peace, and might reconcile them both in one body to God through the cross, by it having put to death the enmity. (Ephesians 2:15-16 NASB)
Colossians: Paul got a little more precise in Colossians 2:14 and 16, where he specified that since the ordinances were nailed to the cross, no one was under obligation to observe the dietary laws, the Sabbaths etc.
having canceled out the certificate of debt consisting of decrees against us, which was hostile to us; and He has taken it out of the way, having nailed it to the cross... Therefore no one is to act as your judge in regard to food or drink or in respect to a festival or a new moon or a Sabbath day-- (Colossians 2:14, 6 NASB)
However the entire issue has been rendered far more complicated by the fact that the verses quoted above seem to flatly contradict statements made by Jesus Himself.
Continue on to Chapter III - Jesus' statements that He had not come to abolish (or destroy) the law or the prophets and that even the tiniest part of the law will not pass away until all things be accomplished, suggests that the binding nature of the law of Moses will remain forever in effect. However, we could find ourselves scratching our heads since the Messiah's words appears to flatly contradict many of the statements made by the New Testament authors. So the million dollar question is - Did Jesus reaffirm/validate Old Testament Law or did He abolish it? HERE
 Daniel B. Wallace. Hebrews: Introduction, Argument, and Outline.
Footnote I. Dating The Book of Hebrews
There is overwhelming evidence, that the three Gospels of Matthew, Mark, and Luke were written within 20-30 years of the Cross, and that the letters of Paul were largely complete before that time.. See Dating The New Testament
When it comes to the book of Hebrews, there is little question that some of the wording clearly indicates that it was written before Jerusalem was destroyed by the Romans in 70 A.D. For example, note the present tense used in the first four verses of chapter 5...
Hebrews 5:1-4: For every high priest taken from among men is appointed on behalf of men in things pertaining to God, in order to offer both gifts and sacrifices for sins; he can deal gently with the ignorant and misguided, since he himself also is beset with weakness; and because of it he is obligated to offer sacrifices for sins, as for the people, so also for himself. And no one takes the honor to himself, but receives it when he is called by God, even as Aaron was.
Hebrews 9:25: nor was it that He would offer Himself often, as the high priest enters the holy place year by year with blood that is not his own. (NASB)
These passages about what high priests do is set entirely in the present tense which would not have been the case if it had been written after 70 A.D. Additionally pay attention to the wording in the the next quote that reads
Hebrews 10:11-12: (11) Every priest stands daily ministering and offering time after time the same sacrifices, which can never take away sins; (12) but He, having offered one sacrifice for sins for all time, sat down at the right hand of God,
The actions of the high priest are in the present tense which means that the rituals they performed were still continuing. However, Jesus' actions are stated to be in the past i.e. "He offered" and "He sat down". In other words, Hebrews was written after Jesus' death, resurrection, and ascension, but before the Temple was destroyed.
Besides which, considering the whole point of Hebrews was to demonstrate Christ's superiority, if the author had known that the Temple no longer existed, he would have used the demise of the sacrificial system to support his argument - something along the lines of commenting that the Old Covenant was so inferior that it had passed away. A foreign army had actually managed to destroy the Temple and, with it, all the rituals that formed the base of Jewish religion.