Kerisos, Chapter 6, Mishnah 5. In the previous Mishnah we established that the obligation to bring a definitive sin-offering or guilt-offering remains in place even after Yom Kippur has passed. This is because, although Yom Kippur atones for the sin, it does not exempt a person from their obligation once a Beis Din has determined that they must bring a sin-offering or the like. In contrast, a suspensive guilt-offering is brought out of doubt, in case the person sinned, and its purpose is to suspend the consequences of the sin. Once Yom Kippur has passed and their sin remains unknown to them, they are purified from their sin and there is no longer a need for a suspensive guilt-offering.
Our Mishnah discusses a case where the obligation for the sin-offering itself is in doubt, and whether one can bring a sin-offering out of doubt. Such a case exists with a bird sin-offering, which is brought in a number of situations, and we will focus on the simplest of them: a woman who miscarried, and it is unknown what the status of the fetus is, whether it had the status of a viable offspring, in which case she is obligated to bring a bird sin-offering at the end of the process, or whether it did not have the status of an offspring, and she is not obligated in it at all.
Why a bird sin-offering can be brought out of doubt:
There is no slaughtering process for a bird - and therefore there is no concern here of slaughtering an unconsecrated animal in the Temple courtyard.
The body of the bird does not go up on the Altar - and there is no concern here of bringing unconsecrated items up to the Altar.
Its consumption - the bird's neck is nipped with a fingernail from the back of the neck and not slaughtered. If it was not consecrated, it requires slaughtering in order to be kosher, and since it was not slaughtered, it cannot be eaten.
Another fundamental principle we prefaced: when a person consecrates an animal for a sin-offering based on a mistaken assumption, thinking that they are obligated in a sin-offering and it turns out they are not obligated, the consecration was done in error and does not take effect, and the animal is not consecrated at all. This is the entire background.
The first part of the Mishnah:
"Ha'ishah sheyesh aleha chatas ha'of safek, she'avar aleha Yom Hakippurim - chayaves lehavi le'achar Yom Hakippurim, mipnei shemachshartah le'echol bezavachim" - A woman who has a doubtful obligation for a bird sin-offering, and Yom Kippur passed over her, is obligated to bring it after Yom Kippur, because it renders her fit to eat from the sacrifices.
A woman upon whom a doubtful obligation for a bird sin-offering took effect - perhaps she miscarried and perhaps it had the status of an offspring - is obligated to bring it even after Yom Kippur has passed. The reason: This sacrifice does not come to atone for a sin, and although it is called a sin-offering, its purpose is purity. Until she brings the bird sin-offering, she is in the category of one who lacks atonement, and she is not permitted to enter the Temple, nor is she permitted to eat consecrated items - for example, they do not allow her to eat from the Passover sacrifice. Therefore, she is obligated to bring the bird out of doubt, so that she will be rendered fit to eat from the sacrifices.
This is the case even according to the opinion of Rabbi Shimon in Tractate Niddah (31), whose words are cited in Rashi on the Torah, who explains what sin a woman who gives birth commits: that she swore out of the pain of childbirth not to be intimate with her husband again. Even he concedes that the point remains the same, and according to his view, as the Gemara says, the pain she went through during the birth itself atones for this anyway. Rather, the role of this sin-offering is not atonement for a sin, but rather purity - to grant her a status of purity - and therefore she is obligated to bring it in any case.
The second part of the Mishnah:
What is the law regarding the bird that was brought to the Temple due to the doubt, when it became clear that she was not obligated in it? If she separated the bird, and before she arrived at the Temple they informed her that the doubt was clarified and there was no status of an offspring here and she is not obligated in a bird sin-offering - it emerges that the separation was done in error, the bird is not consecrated, and she has the right to take it and do with it as she wishes.
But what is the law when "chatas ha'of haba'ah al hasafek... im mishenimlekah noda lah - harei zo tikaver" - a bird sin-offering brought out of doubt... if after its neck was nipped it became known to her, it must be buried? That is, she brought the bird to the Temple based on the possibility that there was the status of an offspring here, and its neck was already nipped - it was killed by cutting with a fingernail from the back of the neck - and only afterward did it become known to her that she was not obligated. The Mishnah states: It requires burial.
Based on the logic we established, it retroactively turns out that she consecrated it by mistake, since she was not actually obligated to bring a chatas offering, and the bird is therefore chullin. Since it died without being properly slaughtered, it is a neveilah, and therefore forbidden to be eaten - but it is seemingly not forbidden to derive benefit from it, and she should theoretically be able to take it home and do with it as she pleases.
Nevertheless, the Mishnah states that it must be buried, and this is a Rabbinic law: The Sages were concerned that onlookers might be led to make a mistake when they see sacrificial birds that had their necks nipped and are being taken to the owners' homes, and they would misunderstand the laws of invalid consecrated items. Therefore, it was established that they must be buried, so that a person does not come to derive benefit from them and misunderstand the halachah - and as mentioned, this is purely a Rabbinic law.
In summary: In this Mishnah, we learned two laws regarding a bird chatas brought out of doubt. The first: A woman who is obligated to bring a bird chatas out of doubt and Yom Kippur passes - she must still bring it, because the offering does not come to atone for a sin, but rather to permit her to eat sacrificial meat, and even Rabbi Shimon agrees with this. The second: If she found out after the neck was nipped that she was not obligated to bring the offering, even though strictly speaking the mistaken consecration is void and the bird is chullin and a neveilah, the Sages decreed that it must be buried, out of concern that people might misunderstand the halachah and come to derive benefit from it.