Gemara

Eating Kodshim is a need of the Avodah:

Beraisa: "V'Ochlu Osam Asher Kupar Bahem" - this teaches that Kohanim eat and the owner of the Korban gets Kaparah.

Yevamos 90a - Beraisa: If Dam became Tamei and it was thrown on the Mizbe'ach:

The Korban is Meratzeh if it was b'Shogeg, but not if it was b'Mezid.

Beraisa: The Tzitz is Meratzeh for Dam, meat and Chelev that became Tamei, whether Mezid or Shogeg, whether willingly or unwillingly, for an individual or the Tzibur.

Question: Mid'Oraisa, in either case it is Meratzeh. Chachamim decreed that b'Mezid it is not Meratzeh. The Korban brought in its place is Chulin ba'Azarah!

Answer (R. Yosi bar Chanina): It was not Meratzeh to permit eating the meat, but it was Mechaper (no other Korban is brought).

Question: In any case, the Mitzvah of eating the meat is uprooted!

"V'Ochlu Osam Asher Kupar Bahem" - Kohanim eat and the owners receive Kaparah.

Pesachim 59a - Beraisa: The only Avodos after the afternoon Tamid are the Ketores, Menorah, Korban Pesach and (the Korban of) a Mechusar Kipurim on Erev Pesach. The Mechusar Kipurim immerses (after his Korban is brought) and eats the Pesach at night.

R. Yishmael, son of R. Yochanan ben Brokah says, also Mechusar Kipurim on any day may follow the Tamid. He immerses and eats Kodshim at night.

Question: Granted, the first Tana holds that (the Korban brought to enable fulfilling) the Mitzvas Aseh of Korban Pesach, which has Kares (for one who does not bring it) overrides the Aseh of Hashlamah (not to offer Korbanos after the afternoon Tamid), which does not have Kares.

But according to R. Yishmael, why does one Aseh (eating other Kodshim) override the other Aseh (Hashlamah)? (Neither has Kares!)

Pesachim 121a - Mishnah - R. Yishmael: If one blessed on the Pesach, he need not bless on the Zevach (Chagigah of the 14th). If he blessed for the Zevach, he must also bless for Pesach.

R. Akiva says, neither of them exempts the other.

Gemara: R. Yishmael considers Shefichah (pouring Dam on the Mizbe'ach) to be (a form of) Zerikah (Birkas ha'Pesach exempts Birkas ha'Zevach because if one poured Dam Zevach like a Pesach, he was Yotzei). However, Zerikah is not a form of Shefichah;

R. Akiva holds that neither is a form of the other.

Tosefta (Sof Pesachim): Birkas ha'Pesach is 'Asher Kidshanu b'Mitzvosav v'Tzivanu Le'echol Es ha'Pesach'. Birkas ha'Zevach is '...Le'echol Es ha'Zevach'.

Zevachim 97b - Question: If Kodshim became forbidden by touching Pasul Kodshim, the Mitzvas Aseh to eat Kodshim should override the Lav (of eating Pasul Kodshim)!

Answer #1 (Rava): An Aseh does not override a Lav pertaining to the Mikdash;

The Aseh to eat Besar Pesach includes marrow, but it does not override the Lav of breaking bones of Pesach!

Answer #2 (Rav Ashi): "Yikdash" is an Aseh (to treat what touches Kodshim like the Kodshim). The Ase to eat Kodshim does not override a Lav and (this) Ase.

Rishonim

Rambam (Hilchos Ma'aseh ha'Korbanos 10:1 and Sefer ha'Mitzvos Aseh 89): Eating Chatas and Asham is a Mitzvas Aseh - "V'Ochlu Osam Asher Kupar Bahem". The Kohanim eat and the owner gets Kaparah. Other Kodshim that Kohanim eat is also a Mitzvah.

Question: Why did the Rambam specify Chatas and Asham? It is also a Mitzvah to eat other Kodshim!

Answer (Radvaz): The primary Mitzvah is eating Chatas and Asham, which come for Kaparah. Regarding Kodshim Kalim and Terumah there is an aspect of a Mitzvah.

Rambam (Sefer ha'Mitzvos ibid.): Eating other Kodshim, and also Terumah, is drawn after the Mitzvah. It is not the same as eating Chatas and Asham, for in these Achilas Kohanim completes the Kaparah. The Sifra equates eating Kodshim Kalim to eating Kodshei Kodoshim regarding Kidush (washing) hands and feet.

Ramban (Mitzvas Aseh Rishonah that the Rambam omitted): Also eating Ma'aser Sheni and all Kodshim is an Aseh. One opinion says that this Aseh overrides the Aseh of Hashlamah (Pesachim 59a).

Rambam (Hilchos Chametz u'Matzah 8:7): He blesses 'Asher Kidshanu b'Mitzvosav v'Tzivanu Al Achilas ha'Zevach' and eats from the Chagigah. He blesses '...Le'echol Es ha'Pesach' and eats from it. Neither Berachah exempts the other.

Rashi (Berachos 48b DH Ki Hu): "He will bless the Zevach and afterwards the people will eat". He blesses 'Asher Kidshanu b'Mitzvosav v'Tzivanu Le'echol Es ha'Zevach'. The command to eat Zevachim is "...Veha'Basar Tochel" (Devarim 12 27).

Rashi (Pesachim 59a DH bi'She'ar): The Mitzvah of eating Kodshim applies to Yisraelim as well as to Kohanim.

Rashi (Sanhedrin 70b DH l'Ituyei): We might have thought that one can become a Ben Sorer u'Moreh by eating the meal to console mourners, for it is only a Rabbinic enactment. We would say that the Chaburas Mitzvah (for which one cannot become a Ben Sorer u'Moreh) refers to eating Pesachim or Kohanim eating Kodshim.

Yere'im (95, cited in Hagahos to Minchas Chinuch (Machon Yerushalayim) 102:2): This shows that Rashi holds that it is not a Mitzvah (mid'Oraisa) for Yisraelim to eat Kodshim other than Pesach.

Rebuttal (Einayim l'Mishpat Berachos 48b Rashi DH Ki): Rashi explicitly says that it is a Mitzvah in Berachos and Pesachim. In Sanhedrin Rashi mentions Kodshim that were normally eaten in a Chaburah.

Rosh (Nedarim 4b DH Ka Sharya): (Also for women) eating Kodshim is a Mitzvah. Also, her eating helps avoid Nosar.

Ran (Nedarim 4b DH ka'Sharya): Eating Kodshim is a Mitzvah, e.g. Pesachim. A woman is also obligated.

Sefer ha'Chinuch (102): Kohanim are commanded to eat some Kodshim, e.g. Chatas, Asham, and parts of Kodshim Kalim given to Kohanim. Their eating completes the Kaparah. However, when they eat (the rest of) Kodshim Kalim or Terumah this does not benefit the one who gave the Terumah or Kodshim.

Acharonim

Kuntres Acharon of Sha'agas Aryeh (33 b'Sof DH Ach): The Rambam includes the Mitzvah of eating Korbanos in the Mitzvah of offering each Korban properly. He does not count it as a separate Mitzvah because it does not have its own Tzivuy.

Tzelach (Beitzah 19b DH u'Midei): Tosfos (Zevachim 97b DH Neisi) holds that all Achilas Kodshim is an Aseh. The Rambam seems to hold that eating Kodshim is an Aseh only regarding Pesach and for Kohanim to eat Chatas and Asham. However, in Hilchos Chametz u'Matzah he says that we bless 'Asher Kidshanu b'Mitzvosav' on eating Chagigah of the 14th! Perhaps he holds that it is a Mitzvah, but he does not count it among the 613 according to his criteria. I am unsure whether or not it is an Aseh that can override a Lav.

Einayim l'Mishpat (ibid.): Perhaps only Chagigah is a Mitzvah, in order that Pesach will be eaten when satiated. Riva (Tosfos Pesachim 70a DH Lav) says that Chagigah of the 14th is only mid'Rabanan.

Note: Even if the imperative to bring Chagigah is mid'Rabanan, one who brings it fulfills a Mitzvah mid'Oraisa. Perhaps eating it is also mid'Oraisa! If it is only mid'Rabanan, obviously its Berachah could not exempt Pesach. How could the Gemara infer that R. Akiva holds that neither exempts the other because Zerikah is not Shefichah?

Objection (Minchas Chinuch DH v'Hinei b'Rambam): The Rambam and Chinuch did not specify whether or not eating other Kodshim is a Mitzvah. However, the Gemara (Pesachim 59a) and Rashi there clearly show that it is! There is no reason for the Tzelach to be unsure.

Answer (Hagahos Tzelach ha'Shalem 19b (8), in the name of Ba'al Minchas Chinuch): The Rosh and Ran (Nedarim 4b) argue about whether or not Achillas Kodshim Kalim other than Pesach is a Mitzvah.

Einayim l'Mishpat (ibid.): If eating Kedushah is itself a Mitzvah, the Shi'ur is a k'Zayis, and it must be eaten normally, and l'Moshchah (in grandeur).