Gemara

43b (Rav Chisda): A Kohen may be Mitamei for a Mes Mitzvah even if it is not a relative, all the more so for his father (even if the body is incomplete)!

Question (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.

Since the son of the Mes is here, this is not a Mes Mitzvah (he must hire others to bury him)!

Answer: The Mes is in a valley (and others are not around), so it is as if there is no one else to bury him.

Yevamos 89a (Beraisa): If a man was married (mid'Rabanan) to a minor, he becomes Tamei to engage in her burial (even if he is a Kohen).

This is not considered uprooting the Torah. 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 her.

Rishonim

Rif and Rosh (Hilchos Tum'ah (after Menachos). 1a and 2): A Mes Mitzvah is one with no one to bury it. If he calls and people answer, it is not a Mes Mitzvah.

Rambam (Hilchos Evel 3:8): A Mes Mitzvah is a Yisrael cast on the road with no one to bury it. This is a tradition (that a Kohen becomes Tamei for it). It is not a Mes Mitzvah if the Kohen could call and others would come and engage in it.

Question (Mizrachi Emor 21:1 DH Lo): The Rambam holds that all the verses are Asmachtos. The Gemara asked why we need a verse to permit Tum'as Nazir for a Mes Mitzvah. We should be able to learn from a Kal va'Chomer from a Kohen Gadol! According to the Rambam, this is not a question!

Answer (Lechem Mishneh): The Rambam wrote in Hilchos Kil'ayim (10:29) that it is a proper Drashah. He calls everything not explicit in the Torah 'Divrei Kabalah.' Alternatively, he means that it is a Gezeras ha'Kasuv.

Rambam (Hilchos Kil'ayim 10:29): Kavod ha'Briyos does not override a Lav of the Torah. A Mes Mitzvah overrides (Tum'as Nazir or Kohen), for it says "ul'Achoso."

Sefer ha'Mitzvos (Aseh 231): It is a Mitzvah to bury a dead Yisrael that same day. This is why a Mes with no one to bury it is called a 'Mes Mitzvah.'

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.

Rosh (Hilchos Tum'ah Siman 2): Yevamos 89a says 'since her relatives do not inherit her, if she calls, no one will answer her.' I.e. if the Mes were alive and he would call, and his relatives would not come for him, he is a Mes Mitzvah, even if other Yisraelim would come. The Ramban explains that even her family would not answer, all the more so others would not, since she has heirs and a family. The Yerushalmi discusses whether or not people would answer when the Kohen who finds the Mes calls. The text in Yevamos should say similarly. It was corrupted by scribes who had not learned the Yerushalmi.

Nimukei Yosef (Yevamos 28b DH uvi'Gmara): Some say that a Mes Mitzvah is when no one answers when he calls, i.e. the person about to die has no relatives. This is even if there are others who would bury him. Even a Kohen or Nazir may bury him, like it says here; since her relatives do not inherit her, they would not answer her, so she is a Mes Mitzvah. This is wrong. A Kohen may become Tamei for a Mes Mitzvah due to Kavod ha'Briyos (to avoid disgrace), so the Mes will not remain unburied. If others could bury it, there is no disgrace! Even if a Kohen who finds the Mes could hire people to bury it, he may choose to become Tamei.

Poskim

Shulchan Aruch (YD 374:1): A Kohen is Mitamei for a Mes Mitzvah. Even if he was heading to circumcise his son, he is Mitamei for the Mes.

Shulchan Aruch (3): A Mes Mitzvah is one found on the road or in a city of Nochrim with no one to bury it, and he cannot call for Yisraelim to come, take care of it and bury it. He may not go away and abandon the Mes, even to go to the city to bring people to bury it. Rather, he is Metamei himself and buries it. If he could call and Yisraelim would come, he does so, and they bury it. If they came and there are enough, he refrains.

Source: Ramban and Rosh (ibid.), cited in Beis Yosef DH veha'Ramban.

Beis Yosef (DH v'Eizehu): R. Yerucham says that if the Mes would call in his lifetime and Yisraelim would not answer, this is a Mes Mitzvah. Some say that whenever his relatives would not come, this is called a Mes Mitzvah.

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.

Rema: Some say that if he does not find people to bury it for free, only for money, he need not spend his money. He may be Mitamei.

Gilyon Maharsha: See the Rema YD 157:1 and OC 656:1.

Note: The Rema obligates forfeiting all one's money to avoid transgressing a Lav. Here, he does not require spending money to avoid Dichuy, not an Aveirah. The Gilyon Maharsha (157 DH Tzarich) says that the Rema does not even obligate forfeiting one's money to avoid transgressing passively!

Rema (ibid.): If we are unsure whether the Mes is a Yisrael or Nochri, we follow the majority that are found there, if we cannot say that the Safek is Kavu'a. (If the doubt began in a place where the status is clear, it is like an even Safek.)

Shach (2): We follow the majority of passersby or the majority of cities. If most are Nochrim, one need not bury the body. We do not follow the majority of the closest city, even if it is so close that surely the Mes is from there.

Shach (3): A case of Kavu'a is when there was a group of nine Nochrim and one Yisrael, and one of them was killed there. If it is an even Safek, one must bury it, but a Kohen may not be Mitamei.

Pischei Teshuvah (1): A case occurred in which a circumcised Mes was found with an idol hanging from his neck, like Nochrim. They buried him among Yisraelim. The Chasam Sofer (341) said that it was proper to bury him. However, a tradition from Sinai forbids burying a Rasha among Tzadikim. B'Di'eved, he should not be re-interred elsewhere, for it is a Safek.