Did you ever see a side of beef? If you brought a cow or a bull as a sacrifice, you might think, “Look at the size of that thing! I’m going to be eating korban casserole for a month!” No worries – sacrifices had time limits as to how long they might be eaten. After that, they were called nosar (leftover) and were forbidden to be eaten. Instead, they had to be burned.
We’ve already seen the other side of this mitzvah - way back in Mitzvah #8, the prohibition against leaving over from the korban Pesach. Here, the Torah adds a positive mitzvah to that negative one.
The reason that nosar is prohibited and must be burned is simple: meat goes bad. After a relatively short amount of time, it starts to stink and is generally offensive. It’s not appropriate for someone to be repulsed by a sacrifice, which is holy. To avoid this, sacrifices have time limits, after which they must be burned.
You will note that God commanded that the leftover meat be utterly destroyed, not put to some other use, such as being used to feed one’s dogs. From this, the Sefer HaChinuch derives a lesson about putting our trust in God. Here, God tells us to rely upon Him and burn this meat. Similarly, one should not be a miser or a hoarder. Rather, we should trust that God has all of our needs under control. (This a very important concept in Judaism with numerous Biblical verses to this effect – see Psalms 145:16, 136:25, 37:25, et al.)
The obligation to burn nosar includes within it any sacrifices gone wrong. If a sacrifice became disqualified for whatever reason, it was immediately burned. (In a case of doubtful disqualification, the sacrifice in question was left until it became nosar and then burned.)
This mitzvah applies to male kohanim in Temple times. It is discussed in the Talmud in tractates Pesachim (27b), Temurah (34a) and Chulin (90b, 125a). It is codified in the Mishneh Torah in the eighteenth and nineteenth chapters of Hilchos Pesulei HaMukdashin. It is #91 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos.