Pesachim Chapter 7, Mishnah 4. As we have learned, the Pesach offering is brought even when the people are tamei: if most of Klal Yisrael are tamei, or if most of the kohanim are tamei, the Pesach offering is brought in a state of tumah. Our Mishnah lists five public offerings that are certainly brought on Shabbos, and are also brought in tumah. Although they are among the offerings that are eaten, they are not eaten in tumah - unlike the Pesach offering.
The words of the Mishnah: "Chamishah devarim ba'in b'tumah v'einan ne'echalin b'tumah" - five offerings are brought even when Klal Yisrael are tamei or the kohanim are tamei, but in such a situation they are not eaten.
And these are the five offerings:
Ha'omer - the barley meal offering brought on the second day of Pesach.
Shtei ha'lechem - the two loaves brought on Shavuos from wheat, the first meal offerings from the year's new grain.
Lechem ha'panim - the showbread on the table, which is set out anew every Shabbos.
Zivchei shalmei tzibbur - the animals brought together with the two loaves on Shavuos.
Se'irei roshei chodashim - the goats brought as a public sin offering on Rosh Chodesh. And although the Mishnah mentions only Rosh Chodesh, the intent includes the other festivals as well, on each of which a goat is brought as a sin offering.
What is unique about these offerings? Surely there are many other offerings that are brought in tumah, but these are distinctive in that they contain a component of eating, and that component goes unused when they are brought in tumah. The Torah permitted bringing them in tumah, but did not permit eating them.
The reason for this: the Torah established that these offerings must be brought at their specific appointed time, and therefore they are brought even on Shabbos and even in tumah - but that is no license to eat them. For the eating is not essential to these offerings, and the offering is entirely valid without it. There is a mitzvah to eat them, and the kohanim do eat all of them, but the eating does not affect the fitness and validity of the offering.
The Pesach that comes in tumah:
The Mishnah continues: "Ha'Pesach she'ba b'tumah ne'echal b'tumah" - unlike the preceding offerings, a Pesach that is brought when most of Klal Yisrael are tamei is also eaten in tumah. The reason: from the outset it comes for the purpose of being eaten. Already in its first description in the Torah, in Parashas Bo, it says "ish lefi ochlo" - each person according to how much he can eat.
It emerges that the eating, the ability to eat the Pesach offering, is essential to its very definition. As we saw earlier, one who slaughters a Pesach offering on behalf of someone who cannot eat - it is not a valid Pesach offering, and the mitzvah of the Pesach offering is not fulfilled through it. And since the eating is essential, the eating is consequently permitted even in tumah.
In summary: five public offerings are brought in tumah but are not eaten in tumah - the omer, the two loaves, the showbread, the public shelamim offerings and the goats of Rosh Chodesh - because eating them is a mitzvah that is not indispensable. In contrast, a Pesach offering that comes in tumah is also eaten in tumah, since the eating is part of the very essence of the offering and is indispensable to it.