When and how were the temple sacrifices stopped after Jesus introduced the sacrament?
We know that, right after His crucifixion, the Lord excused His followers from the necessity of blood sacrifices. See, for example, 3 Nephi 9:19:
And ye shall offer up unto me no more the shedding of blood; yea, your sacrifices and your burnt offerings shall be done away, for I will accept none of your sacrifices and your burnt offerings.
We presume that such sacrifices among the Nephites stopped immediately. But in Judea, such sacrifices continued in the temple as they had for the last thousand years. Early Christian converts up until Paul’s ministry were nearly all Jews, and it seems that many of them continued to observe Jewish temple rites and even sacrifices. See, for example, Acts 21:26, where Paul brings other Christians into the temple to symbolize their ritual purification and even intends that an (animal) offering be made for them.
The temple at Jerusalem was destroyed in AD 70 by Roman legions who were quelling a Jewish revolt. Six hundred years later, rulers sympathetic to Judaism permitted the resumption of animal sacrifices on the temple mount and the reconstruction of the temple; these attempts were ended when Christians briefly regained control of the area. By the end of the seventh century AD Muslims invaders had taken Jerusalem and built the Dome of the Rock on the traditional site of the temple; which structure still stands today.
As for the Church: Joseph Smith observed in passing that animal sacrifice would be briefly resumed as part of the restoration of all things (see generally History of the Church 4:211-212). A church member named Wandle Mace remembered Joseph discussing with the Twelve, the possibility of an animal sacrifice being offered at the Kirtland temple; though it is unclear whether this sacrifice was ever done. Wilford Woodruff also recalled Brigham Young describing plans for a space for animal sacrifices in the Salt Lake Temple; though no such facility was ever actually incorporated in the finished temple. However, Joseph Fielding Smith has affirmed:
Now in the nature of things, the law of sacrifice will have to be restored, or all things which were decreed by the Lord would not be restored. It will be necessary, therefore, for the sons of Levi, who offered the blood sacrifices anciently in Israel, to offer such a sacrifice again to round out and complete this ordinance in this dispensation. Sacrifice by the shedding of blood was instituted in the days of Adam and of necessity will have to be restored.
The sacrifice of animals will be done to complete the restoration when the temple spoken of is built; at the beginning of the millennium, or in the restoration, blood sacrifices will be performed long enough to complete the fulness of the restoration in this dispensation. Afterwards sacrifice will be of some other character. (Doctrines of Salvation 3:94)