Offerings and sacrifices, war, slavery, holy lying, polygamy, concubines, tithing, holy days, feasts, meat laws, musical instruments for worship, jewelery, capital punishment, revenge, and oaths were put away with the old covenant.
Following are some examples with new covenant and old covenant references.
Divorce and Remarriage — Under the Mosaic law you could put away wife if you “found some uncleanness in her” (Deut 24:1) and both could remarry. In the New Covenant if a person remarries after they put away their spouse or someone else marries a put away spouse they commit adultery.
“It hath been said, Whosoever shall put away his wife, let him give her a writing of divorcement: But I say unto you, That whosoever shall put away his wife, saving for the cause of fornication, causeth her to commit adultery: and whosoever shall marry her that is divorced committeth adultery.” Mat 5:31-32
Adultery — Under the Mosaic law it was the act of adultery now it is when you look at a person with lustful desires to commit adultery with them.
Matt 5:27-28 Ye have heard that it was said by them of old time, Thou shalt not commit adultery: But I say unto you, That whosoever looketh on a woman to lust after her hath committed adultery with her already in his heart.
Oaths — Under the Old Covenant you were to make oaths by God’s name (Deut 6:13). Under the New Covenant you are not to make any oaths.
5:33-34 Again, ye have heard that it hath been said by them of old time, Thou shalt not forswear thyself, but shalt perform unto the Lord thine oaths: But I say unto you, Swear not at all; neither by heaven; for it is God’s throne:
Non-Resistance — Under the Old Covenant it was an eye for an eye (Ex 21:24). Under the New Covenant you are not to resist evil and when some one smites you to turn the other cheek.
“Ye have heard that it hath been said, An eye for an eye, and a tooth for a tooth: But I say unto you, That ye resist not evil: but whosoever shall smite thee on thy right cheek, turn to him the other also.” Mat 5:38-39
Love Your Enemy — While under the Old Covenant there is no commandment to hate your enemies but at the same time there is no commandment to love your enemies. Under the New Covenant we are told to love, bless, do good, and pray for our enemies.
Ye have heard that it hath been said, Thou shalt love thy neighbour, and hate thine enemy. But I say unto you, Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which despitefully use you, and persecute you; Mat 5:43-44.