In Matthew Jesus states that He did not come to abolish the Law and the Prophets but to fulfill them; Hebrews discusses that His coming enabled a new covenant and the old is obsolete. What role does the O.T. play in modern Christians' lives?
First, read the answer to Question 131 which gives a general background on the relationship between the Christian dispensation and the Old Testament.
The additional part of this question relates to the meaning of Christ’s fulfilling the Law (Matthew 5:17-18) and the establishment of a new covenant (Hebrews 9:15-17).
Matthew 5:17-18. The context of this passage is the Sermon on the Mount, the purpose of which was to declare the basic principles of the kingdom and to clarify the relationship of its subjects to the Jewish law, the prophets, and the rabbinical traditions. It deals with a wide range of subjects, including the purpose and nature of the Law of Moses. The Lord makes clear that the Mosaic code was of temporary duration and that it was to give way to the Law of Christ, the first characteristics of which He was then proclaiming. To prevent this teaching from being construed as opposition to the Law of Moses, Christ made clear that his mission was not to destroy the law, but to fulfill it. To abrogate the law before it had fulfilled its purpose would have been to destroy it. To fulfill the law was to do as Jesus did – enable it to achieve the purpose for which it was given. Unfortunately, translations such as Today’s English Version, also known as Good News for Modern Man, makes Jesus say exactly what he denied: “17 “Do not think that I have come to do away with the Law of Moses and the teachings of the prophets. I have not come to do away with them, but to make their teachings come true. Remember that as long as heaven and earth last, not the least point nor the smallest detail of the Law will be done away with—not until the end of all things. 19So then, whoever disobeys even the least important of the commandments and teaches others to do the same, will be least in the Kingdom of heaven. On the other hand, whoever obeys the Law and teaches others to do the same, will be great in the Kingdom of heaven. 20I tell you, then, that you will be able to enter the Kingdom of heaven only if you are more faithful than the teachers of the Law and the Pharisees in doing what God requires.” This contempt for Scripture arises from a greater commitment to denominational doctrine that to scholarship. A prior version of this translation was even worse, asserting that Jesus came to give the Law “true meaning.” Both translations contradict what Jesus said. He did not come to give the law “true meaning”; it possessed “true meaning” from the time it was given. He did not come to “make their teachings come true”; their teaching had always been true. Jesus declared that he came to fulfill the law. The Greek word translated fulfill is plhro,w,,,, which means “to fill, fulfill.” Thayer defines its use in Matthew 5:17 as “universally and absolutely, to fulfill, i.e., to cause God’s will (as made known in the law) to be obeyed as it should be, and God’s promises (given through the prophets) to receive fulfillment: Matt. v. 17.” Jesus did not come to perpetuate the law and he identified the time that it would pass away – when all things were fulfilled. Again, the TEV says that it will not pass away “until the end of all things.” This, of course, is the consummation of the ages. Thus, it become important to identify when all things will be “fulfilled.” Jesus said that he came to fulfill the Law. Was He able to accomplish the purpose for which he came? Absolutely. He obeyed the law perfectly and at Calvary it was nailed to the cross and taken out of the way. See the answer to Question 131 for a discussion of the end of the Law. “Christ is the end of the law for righteousness to every one that believeth.” Romans 10:4.
Hebrews 9:15 -17. The coming of Christ not only enabled a new covenant, it necessitated a new covenant. “12 For the priesthood being changed, there is made of necessity a change also of the law. 13 For he of whom these things are spoken pertaineth to another tribe, of which no man gave attendance at the altar. 14 For it is evident that our Lord sprang out of Juda; of which tribe Moses spake nothing concerning priesthood. 15 And it is yet far more evident: for that after the similitude of Melchisedec there ariseth another priest, 16 Who is made, not after the law of a carnal commandment, but after the power of an endless life. 17 For he testifieth, Thou art a priest for ever after the order of Melchisedec. 18 For there is verily a disannulling of the commandment going before for the weakness and unprofitableness thereof. 19 For the law made nothing perfect, but the bringing in of a better hope did; by the which we draw nigh unto God. {the bringing...: or, it was the bringing in} 20 And inasmuch as not without an oath he was made priest: 21 (For those priests were made without an oath; but this with an oath by him that said unto him, The Lord sware and will not repent, Thou art a priest for ever after the order of Melchisedec:) {without...: or, without swearing of an oath} 22 By so much was Jesus made a surety of a better testament.” Hebrews 7:12-22. For the new covenant or testament to become effective it was necessary for the testator to die. “5 And for this cause he is the mediator of the new testament, that by means of death, for the redemption of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance. 16 For where a testament is, there must also of necessity be the death of the testator. {be: or, be brought in} 17 For a testament is of force after men are dead: otherwise it is of no strength at all while the testator liveth.” Hebrews 9:15-17. Upon the death of Christ and the establishment of the church or kingdom, the New Testament of His blood (Matthew 26:28) began to be proclaimed “for the remission of sins.” “37 Now when they heard this, they were pricked in their heart, and said unto Peter and to the rest of the apostles, Men and brethren, what shall we do? 38 Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost. 39 For the promise is unto you, and to your children, and to all that are afar off, even as many as the Lord our God shall call.” Acts 2:37-39.
You Must Hear the Gospel
Romans 10:17
You must hear the gospel and then understand and recognize that you are lost without Jesus Christ no matter who you are and no matter what your background is. The Bible tells us that “all have sinned, and come short of the glory of God.” (Romans 3:23) Before you can be saved, you must understand that you are lost and that the only way to be saved is by obedience to the gospel of Jesus Christ. (2 Thessalonians 1:8) Jesus said, “I am the way, the truth, and the life: no man cometh unto the Father, but by me.” (John 14:6) “Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved.” (Acts 4:12) "So then faith cometh by hearing, and hearing by the word of God." (Romans 10:17)
You Must Believe
Hebrews 11:6
You must believe and have faith in God because “without faith it is impossible to please him: for he that cometh to God must believe that he is, and that he is a rewarder of them that diligently seek him.” (Hebrews 11:6) But neither belief alone nor faith alone is sufficient to save. (James 2:19; James 2:24; Matthew 7:21)
You Must Repent
Acts 3:19
You must repent of your sins. (Acts 3:19) But repentance alone is not enough. The so-called “Sinner’s Prayer” that you hear so much about today from denominational preachers does not appear anywhere in the Bible. Indeed, nowhere in the Bible was anyone ever told to pray the “Sinner’s Prayer” to be saved. By contrast, there are numerous examples showing that prayer alone does not save. Saul, for example, prayed following his meeting with Jesus on the road to Damascus (Acts 9:11), but Saul was still in his sins when Ananias met him three days later (Acts 22:16). Cornelius prayed to God always, and yet there was something else he needed to do to be saved (Acts 10:2, 6, 33, 48). If prayer alone did not save Saul or Cornelius, prayer alone will not save you. You must obey the gospel. (2 Thess. 1:8)
You Must Confess
Romans 10:9
You must confess that Jesus Christ is the Son of God. (Romans 10:9-10) Note that you do NOT need to make Jesus “Lord of your life.” Why? Because Jesus is already Lord of your life whether or not you have obeyed his gospel. Indeed, we obey him, not to make him Lord, but because he already is Lord. (Acts 2:36) Also, no one in the Bible was ever told to just “accept Jesus as your personal savior.” We must confess that Jesus is the Son of God, but, as with faith and repentance, confession alone does not save. (Matthew 7:21)
You Must Be Baptized
Acts 2:38
Having believed, repented, and confessed that Jesus is the Son of God, you must be baptized for the remission of your sins. (Acts 2:38) It is at this point (and not before) that your sins are forgiven. (Acts 22:16) It is impossible to proclaim the gospel of Jesus Christ without teaching the absolute necessity of baptism for salvation. (Acts 8:35-36; Romans 6:3-4; 1 Peter 3:21) Anyone who responds to the question in Acts 2:37 with an answer that contradicts Acts 2:38 is NOT proclaiming the gospel of Jesus Christ!
You Must Be Faithful Unto Death
Revelation 2:10
Once you are saved, God adds you to his church and writes your name in the Book of Life. (Acts 2:47; Philippians 4:3) To continue in God’s grace, you must continue to serve God faithfully until death. Unless they remain faithful, those who are in God’s grace will fall from grace, and those whose names are in the Book of Life will have their names blotted out of that book. (Revelation 2:10; Revelation 3:5; Galatians 5:4)