Zevachim Chapter 8, Mishnah 5. This Mishnah also deals with parts of animals that became mixed together, but here it speaks of limbs: limbs of an olah offering, which are meant to be burned on the Altar, that became mixed with limbs of a blemished animal. That animal was defective, and therefore the Torah forbids offering its limbs on the Altar.
The basic rule for such a mixture:
Let us assume we have before us three limbs: two from valid olah offerings and one from a blemished animal. On the one hand, there is an obligation to burn the valid limbs on the Altar, and on the other hand, there is a negative commandment against burning the forbidden limb. There are two perspectives here:
According to Torah law: The majority of the mixture is valid, and the minority is nullified by the majority and loses its identity. Accordingly, there would be room to say that the entire mixture should be offered on the Altar, and the minority would lose its status within the majority.
According to Rabbinic law: The forbidden limb is not nullified, because a complete limb is a "significant item" - too significant to be nullified in a mixture.
Consequently, both the obligation to burn the limbs and the prohibition against burning them exist here simultaneously. Therefore, everyone agrees that there is no way to offer them: they are not burned on the Altar, lest a prohibition be violated, but rather we wait until their time has passed, and then they are all burned in the place of burning.
The novelty of the Mishnah - one limb has already gone up:
The Mishnah discusses a case where one of the three limbs has already gone up on the Altar by mistake, as the kohen did not know. Now two limbs remain, and among them is certainly one that must be burned on the Altar, and it is possible that both must be burned - since it is possible that the very limb that went up by mistake was the limb that was not supposed to go up, and everything that remains requires burning on the Altar. What is their law?
The opinion of Rabbi Eliezer:
Rabbi Eliezer says: "Im karav rosh echad meihen - yikrevu kol harashim" - if the head of one of them was offered, all the heads shall be offered, and similarly "kera'o shel echad meihen - yikrevu kol hakera'ayim" - if the leg of one of them was offered, all the legs shall be offered. That is, permission is granted to assume regarding the limb that went up, that it is the one that was not supposed to go up. What is done is done, and as for the two parts that remain - both require burning on the Altar.
It is worth noting carefully: Rabbi Eliezer stated his ruling specifically when two limbs remain. When burning both of them together, a mitzvah is certainly being fulfilled. While it is possible that a transgression is also being committed, the transgression is only by Rabbinic law, since by Torah law the minority is nullified by the majority. And since we can assume that the forbidden limb has already gone up, it is possible that there is not even a Rabbinic prohibition here, and both limbs are entirely a mitzvah. Because of this combination of factors, Rabbi Eliezer is lenient in this scenario.
The opinion of the Sages:
The Sages reject his words and establish: "Afilu karvu kulam chutz me'echad meihen - yeitzei leveis hasereifah" - even if all of them were offered except for one of them, it must go out to the place of burning. Even if two out of three were burned on the Altar, or ninety-nine out of a hundred, and only one limb remains - and even though statistically it makes sense that this is the limb that is obligated to be burned - it may not be brought up. This is because by Rabbinic law the prohibition is not nullified, and thus we have a potential prohibition before us, and we do not fulfill the mitzvah of burning the limbs in a place where there is a concern of committing a transgression. Therefore, the rule is "sit and do nothing": we wait and do nothing until the time for remaining overnight has passed, and then they are all burned.
The Beis HaSereifah - Where is it?
The beis hasereifah is the place where invalid offerings are burned, outside of Jerusalem. The Bartenura states this explicitly in Tractate Yoma, in the context of the burning of the Yom Kippur goat. The Hebrew ArtScroll edition writes that the beis hasereifah in this case is a special, designated place inside the Courtyard, but it is unclear what this means.
Clarifying the Parameters of the Dispute:
Even Rabbi Eliezer agrees regarding both extremes of the dispute:
When only one limb remains: It may not be brought up, lest it is the one forbidden by Rabbinic law. Rabbi Eliezer only permitted it when a pair of limbs remains, since then it is known with certainty that there is a mitzvah to burn at least one of them.
When no limb has been burned yet: When all three limbs are before us in the mixture, even though by Torah law the one is nullified among the three and loses its status, Rabbi Eliezer agrees that they may not be burned, since in fulfilling the mitzvah of burning, a transgression is certainly also committed.
According to this, the wording of the Mishnah "afilu karvu kulam chutz me'echad meihen" - even if they were all offered except for one of them, would be explained according to Rabbi Eliezer: except for one pair - two limbs that can be brought up. For in a case where one limb has already been burned, we can retroactively assume that it was the limb that was not supposed to go up, and permit all the rest: there is nothing more than a Rabbinic issue here, and against it stands the mitzvah of burning.
The Halachah:
The halachah does not follow Rabbi Eliezer, but rather the Sages. One who brings a blemished limb up onto the Altar is liable to lashes, and even in a mixture of one in a hundred it is not nullified by Rabbinic law, because a whole limb is a "significant item" (davar chashuv). Therefore, none of the limbs may be offered, and they all go out to the beis hasereifah to be burned outside of Jerusalem.