For the Jews who rejected Jesus as their Messiah, animal sacrifices done in obedience to the Old Testament covenant were stopped in A.D. 70 at the time of the destruction of Jerusalem and the temple by the armies of Rome. Jesus warned of this in the Gospels.
For the Christian community, animal sacrifices stopped with the death and resurrection of Christ. There were some who were persecuted or pressured by the Jewish community to continue to offer sacrifices either because they rejected Jesus or felt his death was not enough. The book of Hebrews actually deals with this and shows that the Old Testament sacrificial system (the Old Covenant) was temporary until the coming of Christ who was the fulfillment of all that the sacrificial system anticipated. Paul teaches us the same thing in passages like Colossians 2:16f. It was Paul who specifically pointed to Christ as our Passover who was sacrificed for us (see 1 Corinthians 5:7). In keeping with the Lord’s Supper, instituted just before His death, Jesus also celebrated what was actually the last legitimate Passover by which He also pointed to Himself as the sacrifice for our sin.
Today, when Jews observe the Passover they cannot offer sacrifices because sacrifices are only to be offered in Jerusalem and in the temple. Prophetically, many believe that in the future during the time of Daniel’s 70th week, the temple will be rebuilt and sacrifices will again be offered, but only because the Jews continue to reject Jesus as their Messiah. The sacrifices will once again be stopped by the beast of Revelation 13 the Jews will be persecuted until Messiah returns at which time they will turn to Him “whom they have pierced,” to put it in the words of Zechariah.