But when Christ came as high priest of the good things that are now already here, he went through the greater and more perfect tabernacle that is not made with human hands, that is to say, is not a part of this creation. He did not enter by means of the blood of goats and calves; but he entered the Most Holy Place once for all by his own blood, thus obtaining eternal redemption. The blood of goats and bulls and the ashes of a heifer sprinkled on those who are ceremonially unclean sanctify them so that they are outwardly clean. How much more, then, will the blood of Christ, who through the eternal Spirit offered himself unblemished to God, cleanse our consciences from acts that lead to death, so that we may serve the living God! Hebrews 9:11-14

Jesus took care of all that on the cross.

All of the sacrifices in the Old Testament prepared the groundwork for the ultimate sacrifice that was to come. Jesus offered himself as a sinless man on the cross, instead of a unblemished animal without defect on the altar.

He offered himself as the eternal sacrifice, the lamb of God that takes away the sin of the world (as John the Baptist called him when they met). His blood had more value than that of a billion goats, calves or bulls.

If we were to return to sacrificing of animals, we would be saying Jesus did not do enough or his work was incomplete. When Jesus died he said, “It is finished.” No more animal sacrifices needed to happen.

We also don't do any animal sacrifices because the temple in Jerusalem was the only place those could occur.  With the temple currently under Muslim occupation, those sacrifices technically can't occur any where else.