Gemara

Rav Nachman: Tum'as Mes is Hutrah b'Tzibur, i.e. totally permitted, we need not strive to offer Korbanos Tzibur in Taharah;

Rav Sheshes: It is Dechuyah b'Tzibur, i.e. it is permitted only if we cannot offer in Taharah.

Version #1: (The Kohanim are divided into Mishmaros; each Mishmar serves one week, according to a rotation. Each Mishmar is divided into Batei Av; one Beis Av is scheduled to work each day.) If part of a Beis Av is Tamei, all agree that Tehorim do the Avodah. They argue about when the entire Beis Av is Tamei, whether the Avodah is done by Temei'im of the Beis Av or by Tehorim of other Batei Av of the Mishmar.

Version #2: They argue even when part of a Beis Av is Tamei. In all cases Rav Nachman allows Temei'im to do the Avodah.

7a - Question (against Rav Sheshes - Beraisa) Question: "V'Nosa Aharon Es Avon ha'Kodoshim" - what sin does the Tzitz bear?

It cannot be Pigul or Nosar (Korbanos offered with intent to eat Chutz li'Mkomo or Chutz li'Zmano), for it says "Lo Yeratzeh" and "Lo Yechashev" (it is Pasul - even the Tzitz will not help!)

Answer: It bears the sin of Tum'ah, for Tum'ah is Hutrah b'Tzibur.

Answer: Tana'im argue about this:

(Beraisa - R. Shimon): The Tzitz is Meratzeh whether or not the Kohen Gadol is wearing it at the time.

R. Yehudah says, it is Meratzeh only when he is wearing it.

R. Shimon: Yom Kipur disproves you - it is Meratzeh even when he does not wear it (he wears only linen garments while offering in the Kodesh ha'Kodoshim)!

R. Yehudah: No - that Avodah is acceptable b'Tum'ah because Tum'ah is Hutrah b'Tzibur.

Inference: R. Shimon holds that Tum'ah is Dechuyah b'Tzibur.

Rishonim

Rambam (Hilchos Bi'as Mikdash 4:14-15): Tum'ah is not Hutrah b'Tzibur, rather, it is forbidden. However, when it is impossible to do the Avodah b'Taharah, Tum'ah is Dechuyah, and the Tzitz is Machshir the Korban

Source #1 (Ri Korkus): The Rambam rules that Tum'ah is Dechuyah b'Tzibur because the Gemara in Pesachim (77a and 79a) says that R. Yehudah is the only Tana who holds that Tum'ah is Hutrah b'Tzibur, and Rav Chisda and R. Yitzchak hold that Tum'ah is Dechuyah b'Tzibur. Also, the Gemara suggested that other Tana'im argue about whether Tum'ah is Hutrah or Dechuyah, and concluded that all agree that it is Dechuyah.

Source #2 (Kesef Mishneh): Regarding Isurim the Halachah follows Rav Sheshes against Rav Nachman.

Source #3 (Likutei Halachos 3a DH v'Hinei): Normally, the Halachah follows R. Yehudah against R. Shimon. However, here the Halachah follows R. Shimon because R. Chanina Segan ha'Kohanim and R. Yosi (8a) and the Sugya in Pesachim (77) hold like R. Shimon.

Question: R. Yehudah was forced to say that Tum'ah Hutrah b'Tzibur because he holds that the Tzitz is Meratzeh only when the Kohen Gadol is wearing it. How can the Rambam rules like R. Yehudah regarding Ritzuy Tzitz, and like R. Shimon that Tum'ah is Dechuyah b'Tzibur?

Answer (Ri Korkus, ibid. 7-8): Really, saying that the Tzitz is Meratzeh only when the Kohen Gadol wears it does not force one to say that Tum'ah Hutrah b'Tzibur. We find that even Tum'as Gavra (i.e. of people) is Nidcheh b'Tzibur (e.g. Pesach ha'Ba b'Tum'ah), even though the Tzitz does not Meratzeh for Tum'as Gavra! The Gemara answered the truth, that R. Yehudah holds that Tum'ah Hutrah b'Tzibur, but we know this from a different source.

Rambam (ibid. 14): Even if most of the Kohanim of the Beis Av serving today are Temei'im, Tehorim of the Beis Av must do the Avodah. If the entire Beis Av is Tamei, Tehorim of another Beis Av of the Mishmar do the Avodah. If the entire Mishmar is Tamei, Kohanim of other Mishmaros do the Avodah.

Ri Korkus: The Gemara did not discuss when the entire Mishmar is Tamei, but surely Rav Sheshes similarly requires using Tehorim of other Mishmaros.

Me'iri (Bava Kama 110a DH Hayah (2)): A Korban Tzibur overrides Tum'ah if there are no Tehorim in the Mishmar.

Likewise, Mahadura Basra there explains that Rashi (ibid. DH d'Ika) holds that Rav Sheshes agrees that if the entire Mishmar is Tamei, we do not use Tehorim of other Mishmaros. However, Maharsha explains that Rashi holds like the Rambam. Here, the Me'iri (DH Korban) explains like the Rambam.

Rambam (ibid.): If most of the Kohanim in Yerushalayim are Temei'im, they do the Avodah b'Tum'ah.

Question: We should require the minority to do the Avodah b'Taharah!

Answer #1 (Radvaz): The majority is like the whole, so it is as if all the Kohanim are Temei'im. However, if there are Kohanim Tehorim near Yerushalayim who can come to serve, they can join to help comprise a majority of Tehorim.

Answer #2 (Kehilas Yakov Yoma 2 DH uv'Ikar): The law that a Tzibur is not detained also applies to the Tzibur of Kohanim. I.e., Tum'ah does not inhibit them from their Avodah, rather, they may serve b'Tum'ah.

Me'iri (DH Korban): A Korban Tzibur overrides Tum'ah if there are not Tehorim in Yerushalayim to do it.

Source: Presumably the Me'iri learns from Pesachim 79A, which says that Pesach is brought b'Tum'ah if the majority of Yisrael are Temei'im, or if the Kohanim and Klei Shares are Temei'im (regarding Kohanim it does not say 'majority').