I posted this on someone's thread who had asked if the Sabbath is a law that must be obeyed for salvation:
Col. 2:16 should settle the issue of feasts and Sabbaths, but we really do have to deal with verses like Matt. 5:17, which says that Jesus did not come to abolish the Law. We also have to deal with Paul's appeal to the Law in the early part of 1 Cor. 9: "Do I say this as a man, or does the Law not say the same thing?" (v. 8)
The early Christians saw that Jesus spent the rest of Matthew 5 explaining verses 17 and 18. Verse 17 says Jesus will bring the Law to fulness, and verse 18 says that all the Law's prophecies will come to pass down to the smallest jot.
The rest of the chapter explains that Jesus really did "fill up" (Gr. pleroo) the Law. The Law commands us not to commit adultery. Jesus "filled it up" to its real spiritual meaning, that we are not even to lust. The Law commands us to fulfill our vows, but Jesus commands us to fulfill our every word as purely honest people, so that the law of the vow is no longer needed because we always speak the truth.
So the law was not abolished, but brought to fullness. For example, clean animals are those that ruminate/chew the cud and who part the hoof. God doesn't care about food, cattle, or pigs (1 Cor. 6:13; 9:9), but he does care about us, and he cares about the Law of Christ (1 Cor. 9:21; Gal. 6:2; Heb. 7:12). If we are to be clean, then we must ruminate on the Word of God and part from the world.
Examples of this interpretation of the Law in Scripture are 1 Cor. 9:6-12 and Galatians 4 (Sarah and Hagar). I found this teaching in Against Heresies IV:13, written by a missionary and overseer named Irenaeus around AD 185.
Similar uses of the Law can be found in the Letter of Barnabas (easy to find online) and just about every other early Christian writing.
The Sabbath in particular was also not abolished but brought to fullness. The fleshly nation of Israel kept a fleshly rest. Spiritual Israel keeps a spiritual rest, which is a perpetual rest. To quote Justin Martyr (c. AD 150):
The new law requires you to keep perpetual Sabbath, and you [Jews], because you are idle for one day, suppose you are godly, not understanding why this command was given to you. if you eat unleavened bread, you say the will of God has been fulfilled. The Lord our God does not take pleasure in such observances. If there is any perjured person or thief among you, let him cease to be so. If any adulterer, let him repent. Then he has kept the sweet and true Sabbaths of God. (Dialogue with Trypho 12)
Hebrews 4 teaches that we must strive to enter that Sabbath rest, the perpetual and daily rest in Jesus that is available for the people of God now that we have been brought into the New Covenant, where all of us have the Holy Spirit.