Gemara

43b (Beraisa): A Mes Mitzvah is one with no one (else) to bury it. If he calls and people answer, it is not a Mes Mitzvah.

Kesuvos 103b: R. Chiya died before Rebbi.

Objection: R. Chiya said 'the day Rebbi died, Kedushah was Batel.'

Answer: We must switch that. (Rebbi said 'the day R. Chiya died...')

Pesachim 70b - Question: How would someone know in advance that Korban Pesach will be brought b'Tum'ah?

Answer: The case is, the Nasi died. (Everyone will be Mitamei for him.)

Yerushalmi (Chagigah 12a): To read "Va'Yizbechu la'Shem Zevachim; v'Ya'alu Olos l'Macharas ha'Yom" is like Beis Shamai (who forbid Olos on Yom Tov);

If we read without a break, it is even like Beis Hillel. David died on Shavu'os. Yisrael were Onenim, and could not offer (Chagigah or Olas Re'iyah) that day.

Rishonim

Rambam (Hilchos Evel 3:10): If the Nasi died, everyone is Mitamei for him, even Kohanim. They made it like a Mes Mitzvah for everyone, for everyone is obligated to honor him.

Source (Kesef Mishneh): The Yerushalmi records that the day Rebbi died, Kehunah was Batel (Kohanim were Mitamei for him like Yisraelim). It is not honorable for the Nasi to be buried by heirs or relatives. Rather, all are like his relatives, and are Mitamei for him. Also the Bavli (Kesuvos 103b) says that Kehunah was Batel the day Rebbi died. Pesachim 70b supports this.

Shirei Korban (Nazir 33a Sof DH veha'Nasi): The Gemara in Pesachim is not a proof. Perhaps it refers to what the Torah calls 'Nasi', i.e. the king.

Tosfos (Kesuvos 103b DH Oso): R. Chaim Kohen said 'had I been present when R. Tam died, I would have become Tamei for him.' R. Chiya taught that the day Rebbi died, Kedushah, i.e., of Kehunah, was Batel. This is why it was only 'that day.' The Yerushalmi in Berachos explicitly says that when Rebbi died, they said 'there is no Kehunah today'.

Rebuttal (Tosfos): The Yerushalmi permitted only Tum'ah mid'Rabanan.

Hagahos Maimoniyos (4): Avi ha'Ezri was unsure whether or not the Gadol ha'Dor is like the Nasi.

Beis Yosef (YD 374 DH Kosvu ha'Tosfos): R. Chaim Kohen would have been Mitamei for R. Tam. He was sure that the Gadol ha'Dor is like the Nasi!

Note: The Gemara answered that Rebbi said that Kedushah was Batel the day R. Chiya died. Perhaps R. Chaim Kohen explains that Rebbi said so because R. Chiya's deeds made him the Gadol ha'Dor (Kesuvos 103b).

Radvaz (10): The Yerushalmi says that a Kohen refrains from a Mes Mitzvah only if enough people come to carry the bier and two sets of alternates. If he is needed, or we recognize the Mes, or there are not enough people for the Mes' honor (Kesuvos 17a explains how many this is), the Kohen is Mitamei. Why did the Rambam omit this? He wrote that we are Mevatel Talmud Torah when there are not enough for the Mes' honor, but we cannot learn from Bitul Mitzvas Aseh to permit Tum'as Kohanim. Perhaps he relies on what he wrote here, that everyone is Mitamei for the Nasi, for it is like a Mes Mitzvah, and his honor determines that Kohanim be Mitamei.

Rosh (Hilchos Tum'ah Siman 2): 'We recognize the Mes' means that we know that he learned written and oral Torah, for then we require 12,000 people ... If he taught others, there is no limit to the proper number.

Beis Yosef (DH veha'Ramban and DH v'Da): The Ramban says that the Yerushalmi asked whether Tum'as Kohen is permitted for the Nasi when it is not a Mes Mitzvah, i.e. he died in the city, and he has relatives and people to bury him. The Rosh explains that it asked about a Nasi who did not teach others. If so, all Kohanim must be Mitamei for one who taught. We never saw such a practice! Also, if so, Kehunah is Batel on the day of death of anyone who taught! And why did the Yerushalmi ask whether a Talmid may be Mitamei for his Rebbi? Everyone must be! It is difficult to say that we ask about a Rebbi who taught only written Torah, or he taught only to one Talmid.

Tosfos (Chagigah 17a DH Af Atzeres): The Yerushalmi said that Yisrael did not offer on the day David died due to Aninus. It holds that the day was not Shabbos.

Question (Mishneh l'Melech): Everyone is commanded to become Tamei Mes for the Nasi. If so, they could not become Tahor until the seventh day!

Answer #1: They offered Machar, i.e. later, on the seventh day.

Rejection (and Answer #2 - Mishneh l'Melech): One may not enter the Azarah on day seven (he is a Tevul Yom), nor send his Korbanos with a Shali'ach, for he himself must do Semichah! Yisrael were not Mitamei for David, for they were already obligated to offer Chagigah, Re'iyah and Shalmei Simchah.

Sha'ar ha'Melech (Hilchos Ma'aseh ha'Korbanos 3:3 DH v'Od): One may offer Korbanos without Semichah if Semichah cannot be done. How could Yisrael be Mitamei for the Nasi on Erev Pesach? Pesach may be brought b'Tum'ah, but not Chagigah and Re'iyah! We must say that the Nasi died in the morning, before the obligation for these Mitzvos began. (David died on Yom Tov, and Yisrael were already obligated.)

Sho'el u'Meshiv (1:2 179 DH v'Hinei): We concluded (Sukah 25b) that an individual may not be Mitamei for a Mes Mitzvah after he was obligated to bring Pesach because there is Kares for not bringing it. One may become Tamei and not bring Chagigah, Re'iyah and Simchah, for which there is no Kares. Why didn't Yisrael do so? Rather, mid'Rabanan a Nasi is like a Mes Mitzvah, and all are Mitamei. Chachamim could not enact this to uproot Mitzvos that were already obligatory. Perhaps once one was permitted to enter the Mikdash to bring Pesach b'Tum'ah, he could stay until the morrow to offer Chagigah and Simchah.

Note: How can one offer Simchah, which is for the sake of eating? Only Pesach may be eaten b'Tum'ah. Also, how could most of Yisrael spend the entire night in the Azarah, including roasting and eating Korban Pesach? Even if we will say that there were very people, and there was room in the Kodesh part of the chambers around the Azarah, surely one who needs to use the toilet must leave the Azarah, and may not reenter b'Tum'ah!

Poskim

Shulchan Aruch (YD 374:11): Everyone is Mitamei for the Nasi.

Aruch ha'Shulchan (6): The Tur says that if a Chacham dies, everyone is obligated to be Mitamei for him. The same applies to one's Rebbi. The Poskim do not bring the Tur.