The phrase "but I say" or "but I tell you" appears in the Gospels quite a few times. But what does it mean? Let's examine the first time it appears, which is in Matthew 5:22.
21 "You have heard that it was said to the people long ago, 'You shall not murder, and anyone who murders will be subject to judgment.' 22 But I tell you that anyone who is angry with a brother or sister [without cause] will be subject to judgment. Again, anyone who says to a brother or sister, 'Raca,' is answerable to the court. And anyone who says, 'You fool!' will be in danger of the fire of hell. [Matthew 5:21-22; NIV]
It appears that Jesus is quoting from the Law. And indeed, the Law does say, "you shall not murder" [Exodus 20:13]. However, the Law does not say, "anyone who murders will be subject to judgement". It says the one who murders is to be put to death [Numbers 35:16-21]! Is Jesus making the Law harder to follow? Not at all. For the Law says, "love your neighbor as yourself" [Leviticus 19:18] and, "do not bear a grudge against anyone" [Leviticus 19:18]. What Jesus is doing here is linking Leviticus 19:18 with Exodus 20:13. If you break the later, then you've broken the former in your heart. This wouldn't be the only time that Jesus linked two different commands together, so that breaking one means you break the other in your heart. In Matthew 5:27-28, Jesus linked coveting with adultery.
So what about the next example? Here's a verse where Jesus appears to be overriding God's Law.
31 "Furthermore it has been said, 'Whoever divorces his wife, let him give her a certificate of divorce.' 32 But I say to you that whoever divorces his wife for any reason except sexual immorality causes her to commit adultery; and whoever marries a woman who is divorced commits adultery." [Matthew 5:31-32; NIV]
The Law does indeed say that anyone who divorces his wife must give her a certificate of divorce [Deuteronomy 24:1]. But here's the question: Is Jesus forbidding what God has previously allowed? Or is the intent behind the command in question being misused? Remember the Deuteronomy 13 Test. And the fact that Jesus condemned the Pharisees for nullifying the Law of God:
1 Then some Pharisees and teachers of the law came to Jesus from Jerusalem and asked, 2 "Why do your disciples break the tradition of the elders? They don’t wash their hands before they eat!"
3 Jesus replied, "And why do you break the command of God for the sake of your tradition? 4 For God said, 'Honor your father and mother' and 'Anyone who curses their father or mother is to be put to death.' 5 But you say that if anyone declares that what might have been used to help their father or mother is 'devoted to God,' 6 they are not to 'honor their father or mother' with it. Thus you nullify the word of God for the sake of your tradition. [Matthew 15:1-6; NIV]
So according to Jesus himself, anyone who nullifies the Law of God is guilty of sin. This means that it can't be that Jesus is nullifying Deuteronomy 24:1 by forbidding adultery. So it must be that the intent of Deuteronomy 24:1 is being ignored. What's the intent? The word translated as "uncleanness" is the Hebrew word "erva", which means "nakedness" or "shamefulness". If you read through Scripture carefully, you will find that sin is always linked with shame, with the former justifying the latter. In other words, the intent of the command is to allow a man to divorce his wife if he finds out that she has been sexually immoral.
So then, what's with Matthew 19:8, where Jesus appears to be saying that the commands concerning divorce are not of God? If Jesus was doing just that, it means that the inerrancy doctrine is wrong. Also, remember Deuteronomy 13? Moses could not have put that command there if God had not approved of it! Otherwise, Jesus would be guilty of sin by forbidding what God's Law has allowed. The issue here is the intent of the command, which agrees with what Jesus is saying.
Here's another example of Jesus apparently contradicting God's Law. Mind you that Jesus is God, and God does not change (if he does change, then he cannot be eternal).
33 "Again you have heard that it was said to those of old, 'You shall not swear falsely, but shall perform your oaths to the Lord.' 34 But I say to you, do not swear at all: neither by heaven, for it is God’s throne; 35 nor by the earth, for it is His footstool; nor by Jerusalem, for it is the city of the great King. 36 Nor shall you swear by your head, because you cannot make one hair white or black. 37 But let your 'Yes' be 'Yes,' and your 'No,' 'No.' For whatever is more than these is from the evil one." [Matthew 5:33-37; NKJV]
If we take a careful look at the Greek text, you will notice that the "not" in verse 34 is a qualified not. The most likely qualifier is the word "falsely" between "swear" and "at all". What's going on here is that the Pharisees (in typical fashion) are saying, "If you swear by the Lord, you are bound by that oath. But if you swear by other things, then the oath means nothing." Jesus had a few things to say about the Pharisees rules concerning what you may or may not swear falsely by. To paraphrase, "anyone who swears by anything is bound by the oath that they swear."
One more thing. I noticed that the NIV says in verse 37, "All you need to say is simply 'Yes' or 'No'; anything beyond this comes from the evil one." Notice the clear change in meaning between the two translations. The saying, "let your 'yes' be 'yes' and your 'no' be 'no'" means we are to speak truthfully when we swear. But the saying, "all you need to say is simply 'yes' or 'no'; anything further is from the evil one" is a clear violation of Deuteronomy 12:32.
To summarize, when Jesus said, "but I say", he either contradicted a human rule by what he said, or brought to light the intent of God's command, or linked breaking one command (one that cannot be enforced) with breaking another command (one that can be enforced) in your heart. Never did Jesus speak against God's Law. For he gave the Law and told King David through the Holy Spirit that the Law is eternal [Psalm 119:160].