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The distinction between the Old Testament and New is that the Old Testament laws were given to the nation of Israel as part of the covenant between God and the people of Israel. The "eye for an eye" seems harsh to many people, but it was actually a law of restraint. One eye for one eye--NOT two eyes, one arm and a foot for one eye. "Burn for burn"--not "whole villages aflame". The idea was to put the punishment into congruence with the actual offense. Vengeance would most often greatly exceed this standard.
In the New Testament, Jesus is speaking not to the nation of Israel with regard to the covenant, but to individuals. He was also speaking while the entire world was on the brink of the inception of the new covenant.
So, Old Testament Covenant applies to the nation of Israel as long as that covenant is in effect. That means between God and the nation of Israel between the time of Moses and the inception of the new and better covenant--roughly 1600 B.C. to 33 A.D.
Questions like this come up fairly often on this site. People ask, "Should I sacrifice a lamb or not eat pork or not wear clothing with mixed fabrics?" The answer is, "Yes, if you are Jewish and are living between 1600 B.C. and 33 A.D." However, if you are non-Jewish or are alive today, the Old Testament Convent Laws are not binding on you.