These sacrifices have continued at all times, from Adam until Jesus Christ, but whether they should continue subsequently is disputed by some. This is because the ordinances of God have been changed by man.
In these latter days, the Lord God has restored the fullness of the gospel to the earth, and opened up the dispensation of the fullness of times. In this dispensation, the ordinances of God are being established precisely in accordance with those of preceding dispensations. Among these ordinances are sacrifices, which were to be completely restored when the Temple was constructed.
The purpose of this article is to show from the scriptures that sacrifice will be offered in these modern times; that Joseph Smith, a modern prophet, taught that this was about to take place;
It is clear then, that Jesus had to keep this feast as an Israelite, and that this feast of the passover is still supposed to be kept by the generations of Israel. But if sacrifices had all been fulfilled, the feast of the passover could not be kept, for it was a sacrifice. (Exod. 12:26-27):
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The above quotations illustrate that Jesus did teach and practice the ordinances of sacrifice. Since he commanded his saints to follow his teachings and example, they too must continue these ordinances.
The Apostles Continue Sacrifice.
Paul Makes an Offering
In accordance with the teachings of Jesus, the apostles in the early church continued to offer sacrifices upon the altar. There are several different types of sacrifices that were recorded as having been offered. At a time when some people had incorrectly thought that he had been teaching that these customs were done away with, Paul corrects this fallacy by himself sacrificing as an example. (Acts 21:20-24, 26):
Paul Keeps the Feast of First Fruits
Another sacrifice offered by Paul was that connected with Pentecost, also called the feast of first fruits, as it is written (Acts 20:16):
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This feast of first fruits involves the offering up of a sacrifice (Exod. 23:16-19):
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18 Thou shalt not offer the blood of my sacrifice with leavened bread; neither shall the fat of my sacrifice remain until the morning.
Paul's Writings on Sacrifice
The Epistle to the Hebrews also contains evidence that offerings are still supposed to be made, but these writings are complex, and often ambiguous. That offerings could be made, there is no doubt, for it is written, "every high priest is ordained to offer gifts and sacrifices." (Heb. 8:3.)
The writer continues, writing specifically about sacrifices for sin offerings: "For if we sin wilfully after that we have received the knowledge of the truth, there remaineth no more sacrifice for sins." And again, "Now where remission of these is, there is no more offering for sin." (Heb. 10:26, 18.)
The writer continues by showing that there was a purpose for sacrifices, as there is a reason for all of the ordinances. The purpose of sinofferings, for example, was to "sanctify to the purifying of the flesh," (Heb. 9:13), but the sacrifice of Jesus Christ was a more perfect offering, for it purified the spirit. (Heb. 9:8, 9, 14, 23; 10:14.)
And although bread and wine have always been partaken of in remembrance of Jesus Christ (Gen. 14:18-19), thereby purifying the spirits of the saints; and the children of God have always been baptized for the remission of their sins (1 Cor. 10:1-4); yet, it was still "necessary" that the followers of God offer up sacrifices for their sins. (Heb. 9:23.)
Sacrifices Disappear
Having established the fact that sacrifices remained in the New Testament church, an examination of the reasons for their disappearance is necessary. Clement, an early Christian writer, at the end of a long exhortation for the saints to offer their sacrifices according to the appointed times and places, said:
20. The daily sacrifices are not offered everywhere; nor the peace-offerings, nor the sacrifices appointed for sins and transgressions; but only at Jerusalem: nor in any place there, but only at the altar before the temple; that which is offered being first diligently examined by the high priest and the other minister as we before mentioned. (2)
Since the above law was in effect for the Christian church, it seems probable that sacrifices were not discontinued until the destruction of the Temple in A.D. 70. It is likely that after the Temple was destroyed, the doctrine arose that sacrifices would never be restored to the church.
Daniel
Daniel prophesied years earlier, that men would begin to teach the dogma that sacrifices should cease. He said that it "cast down the truth to the ground; and it practiced and prospered." (Dan. 8:12.) But notwithstanding this false doctrine, it is evident that the true saints would be aware that sacrifices were to be reestablished again, for according to the vision of Daniel, after a lengthy period of time the sanctuary would be cleansed, and this purification would occur "at the time of the end," or in these latter days. (Dan. 8:13-14, 17.)
Read more here: http://www.strangite.org/Offering.htm