Did Jesus revoke the "eye for an eye" principle of Biblical law?
This question refers to one of Jesus' statements in his Sermon on the Mount:
38 “You have heard that it was said, ‘An eye for an eye, and a tooth for a tooth.’ 39 But I tell you, don’t resist him who is evil; but whoever strikes you on your right cheek, turn to him the other also. Matthew 5:38-39WEB
Jesus was referring to a principle of Biblical law which was first (directly) mentioned in Exodus:[1]
22 “If men fight and hurt a pregnant woman so that she gives birth prematurely, and yet no harm follows, he shall be surely fined as much as the woman’s husband demands and the judges allow. 23 But if any harm follows, then you must take life for life, 24 eye for eye, tooth for tooth, hand for hand, foot for foot, 25 burning for burning, wound for wound, and bruise for bruise. Exodus 21:22-25WEB
This is also called the talion principle. The word talion comes from the Latin word talio, which means "like" or "such." It refers to the fact that God limits the legal punishment or penalty for intentional or negligent injury to be "like" the actual crime or injury. In the ancient world, within cultures that promoted vengeance, people would be tempted to exact a greater penalty than the original harm done. God both prohibited personal vengeance (Lev. 19:18) and set a limit upon court-appointed retributive harm: "an eye for an eye, ..." set an upper limit, by equivalence.
In practice, the literal equivalent would almost never be carried out. God allowed the penalty to be converted to monetary restitution through negotiation.
Some foolish Christian teachers assert that Jesus was repudiating the Mosaic Law at this point. This is obvious nonsense, because Jesus had begun his sermon with the strongest statement possible upholding the law:
17 “Don’t think that I came to destroy the law or the prophets. I didn’t come to destroy, but to fulfill. 18 For most certainly, I tell you, until heaven and earth pass away, not even one smallest letter or one tiny pen stroke shall in any way pass away from the law, until all things are accomplished. 19 Therefore, whoever shall break one of these least commandments and teach others to do so, shall be called least in the Kingdom of Heaven; but whoever shall do and teach them shall be called great in the Kingdom of Heaven. 20 For I tell you that unless your righteousness exceeds that of the scribes and Pharisees, there is no way you will enter into the Kingdom of Heaven. Matthew 5:17-20WEB
If Jesus were about to teach people against any of the commandments in the Mosaic law, then the Pharisees would immediately have accused him of being a "hypocrite," an epithet which Jesus was constantly using against the Pharisees: Matt. 23:13, 15, 23, 25; Mark 7:6; Luke 11:44.
In fact, Jesus was teaching against a then-current misapplication of Biblical law: that "eye for an eye" justified personal vengeance (i.e. extra-judicial retaliation). R. T. France, in his commentary on Matthew, writes:
Does this teaching contradict or abrogated the law? Rather, insofar as "an eye for an eye" had come to be used to justify personal retaliation, it is simply declared to be irrelevant to personal ethics. The principle of proportionate retribution should not guide us in our relations with others. But then that was not its intention: it was a judicial guideline, not a license to get one's own back.[2]
Jesus is also not suggesting that all forms of "resistance" against evil are now unacceptable. That would also be a repudiation of God's law. Jesus is commanding against armed "resistance" (rebellion) against the Roman rulers and their lackeys, using the same word (Greek: áντιστáναι) that Paul uses in Romans 13:2 when he says essentially the same thing.
The "evil ones" of that day were the Roman rulers and their local military enforcers. Jewish zealots were continually attempting to get their countrymen to rise up and overthrow the Roman oppressors. Jesus taught against this, understanding that the Roman Empire had been put in power over God's people for God's purposes (see the prophecy of Daniel 2). So when the Roman centurion impressed someone to carry their pack for a mile (which they were allowed to do under Roman law), Jesus commanded cooperation. Even though the Romans were "evil", they were the God-appointed rulers up until they were crushed by the kingdom of God (Dan. 2:35,44).
For more information on early Christian subordination to Rome, and whether this applies today, see What is Paul saying about civil government in Romans 13?