Gemara

A Kohen Gadol Mashu'ach (anointed with Shemen ha'Mishchah) has precedence over a Merubah Begadim (one inaugurated only through wearing the additional garments. If they encounter a Mes Mitzvah, the Merubah Begadim is Mitamei).

Yevamos 89b: A Kohen is Mitamei for his mid'Rabanan wife because she is like a Mes Mitzvah.

Question (Beraisa): A Mes Mitzvah is one that has no one to bury it. If he calls and others answer, this is not a Mes Mitzvah.

Answer: Here, since her relatives do not inherit her, if she calls, no one will answer (they will not want to bury her).

Sanhedrin 104a: R. Freida's grandfather found a skull. He buried it (twice, each time) it came up. He realized that it was the skull of Yehoyakim, about whom it says "Kevuras Chamor Yikaver..."

Yerushalmi (Nazir 33a): "Tikberenu" - you must bury the entire Mes. If anything was omitted, nothing was done. This shows that a Mes Mitzvah is only when the head and majority are there.

Rishonim

Rambam (Hilchos Evel 3:8): A Mes Mitzvah is a Yisrael cast on the road with no one to bury him.

Mishneh l'Melech (14:21 DH Asher): A Mes Mitzvah is only when the head and majority are there. Only then one must bury them (Yerushalmi).

Kol Mevaser (1:1 DH Ulam): The case of Yehoyakim's skull proves that if a buried Mes became exposed, it is like a Mes Mitzvah, and all must bury it.

Rebuttal (Yabi'a Omer 3 YD 21 DH v'Hinei): Perhaps R. Freida's grandfather was stringent beyond the letter of the law.

Tosfos (Nazir 47a DH v'Chen): How can a Mashu'ach and a Merubah Begadim be together? After Yom Kipur, the latter is called Kohen she'Ovar! Rather, they found a Mes Mitzvah on Yom Kipur. Even though one may not bury him, one may be Mitamei to move him from the sun to the shade.

Question (Tosfos Yevamos 89b DH Kivan): The Gemara concludes that whenever a man inherits his wife, her family will not bury her, so she is a Mes Mitzvah. This is not always true! If a woman remarried based on a false witness, her first husband inherits her, but may not bury her! And why can't a Kohen Gadol or Nazir be Mitamei for his relatives when there are no other heirs?

Answer #1 (Tosfos): The Gemara means that she is like a Mes Mitzvah, so it does not looks like Chachamim uproot the Torah.

Answer #2 (Aruch l'Ner): The Rambam permits Tum'ah for a Mes Mitzvah only for burial. The Beis Yosef (YD 373) brings two opinions about whether or not Tum'as Kohen for relatives is only for the needs of burial. If the Rambam permits even not for the needs of burial, he can answer that any sole heir may be Mitamei for needs of burial, but sometimes (e.g. a Nazir, or one who remarried...) the absolute Heter of Tum'ah for relatives does not apply.

Ritva (Yevamos 89b DH veha'Tanya): Tum'ah for a Mes Mitzvah is permitted only for Kavod ha'Briyos, lest the Mes remain unburied.

Question (Maharatz Chayos Berachos 20a): How did the Gemara try to prove from a Mes Mitzvah that Kavod ha'Briyos overrides Lavim? Burying the dead is a Mitzvah!

Note: Normally, An Aseh does not override a Lav if both can be fulfilled. Even if the Kohen could go to get a Yisrael to bury the Mes without Dichuy, he is commanded to bury the Mes himself (if no one can hear him) for Kavod ha'Briyos, to avoid the disgrace of leaving the Mes unattended.

Poskim

Shulchan Aruch (YD 374:3): A Mes Mitzvah is one found on the road or in a city of Nochrim with no one to bury it.

Taz (1): It is a Mes Mitzvah unless there are people to carry the bier and bury it.

Bach (DH k'Shem): Whenever a Kohen would not be Mitamei for a relative, e.g. after the coffin cover is sealed or once he separated from it, or if the Mes was a heretic or informer, he is not Mitamei for a Mes Mitzvah.

Shulchan Aruch (OC 526:8): If Yisraelim in a city find out about an unburied Mes four or five days away, even though they will not be able to get there before the end of the Mo'ed, they may travel on Yom Tov Sheni in order to speed the burial.

Source (Beis Yosef DH Nishal): The Rivash (116) forbade leaving the Techum for the sake of a Mes not in front of us. The Rashbatz (1:22) permitted in any case when there is no one closer to bury it, for it is like a Mes Mitzvah.

Bi'ur Halachah (DH Mutar): This is like a Mes Mitzvah. Even though we do not allow Melachos mid'Oraisa such as making an Aron, we should not be stringent about mid'Rabanan laws at the expense of Kavod ha'Briyos.

Rema (10): One may not bury a Nefel on Yom Tov Sheni. We wait until tomorrow.

Source (Hagahos Maimoniyos Hilchos Milah 1:10): Chachamim made Yom Tov Sheni like a weekday regarding burial. This refers to a regular Mes, for which there is disgrace and an Isur to leave it unburied. There is no Mitzvah to bury a Nefel. We find that people cast them into pits.

Rebuttal (Magen Avraham 20): Casting into a pit is burial! The Gemara (Nidah 57a) explains how Kusim justified not burying Nefalim. This implies that Yisraelim bury them! Toras Kohanim expounds that Kohanim may not become Tamei for Nefalim. Surely there is a Mitzvah of burial, for Kohanim may become Tamei only for burial!

Defense (Binyan Tziyon 113 DH Nidrashti): One opinion in Ohalos 16:5 permits removing bones from a pit into which Nefalim were cast to be Metaher it. If there were a Mitzvah to bury them, they would acquire the place! Perhaps Hagahos Maimoniyos holds like the opinion that permits Kohanim to be Mitamei for needs other than burial. However, the Rambam (Evel 1:9) and Rosh (Mo'ed Katan 3:94) rule like a Beraisa in Maseches Semachos says that we do not engage with a Nefel, i.e. one need not bury it.

Yabi'a Omer (3 YD 22 DH Achen): In the Yerushalmi, Tana'im argue about whether or not a Kohen may be Mitamei for Ever Min ha'Chai of his father. We rule that he may, even though a Kohen may be Mitamei only for needs of burial. According to the Magen Avraham, this shows that there is a Mitzvah to bury Ever Min ha'Chai. However, Orchos Chaim (526:8) refuted his proof from Toras Kohanim. We expound that a Kohen may not be Mitamei for Nefalim because there is no Mitzvah to bury them!