Gemara

A (Kosher) Kohen never becomes profaned.

Sanhedrin 52a (Beraisa - R. Meir): "Es Aviha Hi Mechaleles" - this teaches that we no longer treat her father honorably (like other Kohanim). If we used to treat him like Kodesh, we treat him like Chulin. If we used honor him, we disgrace him.

Bechoros 45b (Mishnah): If a Kohen married a woman forbidden to him, he is Pasul until he vows not to benefit from her. If a Kohen becomes Tamei Mes, he is Pasul until he accepts not to do so again.

Menachos 18b (Beraisa - R. Shimon): "Mi'Benei Aharon" - if a Kohen does not admit to all the Avodos of Kohanim, he has no share in Kehunah.

109a (Mishnah): Kohanim who served in Beis Chonyo may not serve in the Mikdash in Yerushalayim. We need not say that idolatry disqualifies.

"Ach Lo Ya'alu Kohanei ha'Bamos El Mizbach Hash-m bi'Yrushalayim Ki Im Achlu Matzos b'Soch Acheihem" - they are like Ba'alei Mumim. They receive portions and may eat Kodshim, but they may not serve.

Berachos 32b (R. Yochanan): A Kohen who killed someone may not give Birkas Kohanim -- "Yedeichem Damim Male'u."

Rishonim

The Rif and Rosh (Sanhedrin 16a and 7:1) bring the Gemara in Sanhedrin.

Rambam (Hilchos Tefilah 15:3): If a Kohen killed someone, even b'Shogeg, even if he did Teshuvah, he cannot Duchan. The same applies if he served idolatry or became an apostate. It says "Ach Lo Ya'alu..."; Birkas Kohanim is like Avodah. Other Aveiros do not disqualify.

Rambam (6): If a Kohen has nothing that disqualifies from Birkas Kohanim, even if he is not a Chacham and is not meticulous about Mitzvos, or people murmur about him, or his business dealings are unjust, he gives Birkas Kohanim. It is a Mitzvas Aseh for every Kohen able to do so. We do not tell a Rasha to be a bigger Rasha and refrain from Mitzvos.

Rambam (Hilchos Bi'as Makdish 6:9): If a Kohen married forbidden women, he may not serve until Beis Din makes him vow Al Da'as Rabim, which cannot be annulled, that he will not sin again. He serves, descends and divorces. Similarly, if he was Mitamei to Mesim, he is disqualified until he accepts in Beis Din not to do so.

Rosh (Megilah 3:19): Sefer ha'Zahir says that a Kohen who was an apostate and repented may not give Birkas Kohanim or read first in the Torah. "V'Kidashto" - once he profaned his Kedushah, he does not return. A Kohen who served in Beis Chonyo is Pasul for Avodah, like a Ba'al Mum. Rashi says that he can Duchan; a Ba'al Mum may Duchan unless people will look at the Mum. The Pesul is only for Avodas ha'Mikdash in Yerushalayim, but not for Avodah in the Mikdash in Nov or Giv'on, and all the more so not for Birkas Kohanim and reading first in the Torah.

Sefer ha'Chinuch (269): A Kohen who does Aveiros withholds himself from honor.

Minchas Chinuch (DH v'Hinei Gam): One could say that we do not honor a Kohen only if he transgresses special Mitzvos of Kehunah, e.g. marrying a divorcee or becoming Tamei. However, it is reasonable that the same applies to any Aveirah. This is no greater than Kivud Av v'Em. One need not honor a parent who does not act like your nation.

Rivash (94): If a Kohen does not conduct with Kedushas Kehunah, e.g. he marries a Pasul woman, becomes Tamei Mes, or killed someone, even b'Shogeg, or does not admit to all Avodos of Kehunah, he forfeits the Kedushah. Chachamim did not refrain from cursing Kohanim of the greatest lineage who acted improperly (Menachos 53b, Yoma 71b). All the more so this applies nowadays, when Kohanim cannot prove their lineage.

Poskim

Shulchan Aruch (OC 128:39): If a Kohen does not have something that disqualifies from Birkas Kohanim, even if he is not meticulous about Mitzvos and everyone murmurs about him, he may Duchan.

Magen Avraham (56): Ra'anach says that this is only if he did Teshuvah.

Mishnah Berurah (142-144): Aveiros in this Siman disqualify (i.e. marrying women forbidden specifically to Kohanim, Tum'ah, murder and idolatry). Other Aveiros, even severe Mitzvos like Arayos, do not disqualify, even if it is known that he transgresses.

Kaf ha'Chayim (232): He is not disqualified due to rumors of any Aveiros.

Rema: Other Aveiros do not disqualify from Birkas Kohanim.

Gra (75): The Rema says so even if he did not do Teshuvah (Sa'if 35).

Shulchan Aruch (40): If a Kohen married a divorcee, he may not Duchan, and we do not treat him with Kedushah, even to read first in the Torah. Even if he divorced her or she died, he is Pasul until he vows not to benefit from women forbidden to him.

Magen Avraham (57): He is disqualified also if he married a Chalutzah, but not for being ill-mannered and brazen.

Magen Avraham (59): Keneses ha'Gedolah says that Re'em disqualifies before he divorces her, even after he vowed Hana'ah. Why is this different than Avodah? He may serve after vowing, and then divorces her (Gitin 35b)!

Kaf ha'Chayim (239): The Magen Avraham did not see the Re'em. The Re'em forbids only when he will not be able to divorce her immediately.

Shulchan Aruch (41): If he became Tamei to a Mes other than his seven relatives, he is Pasul for Birkas Kohanim and all attributes of Kehunah until he accepts not to do so again.

Mishnah Berurah (151): He must accept in Beis Din. He need not vow, for the Yetzer Ra for this is not as strong as for Arayos. A doctor who profits by inspecting Mesim must vow.

Bi'ur Halachah (DH Nitma): The Mishnah (he becomes Tamei through Mesim) connotes that he does so regularly. The Shulchan Aruch connotes that he is disqualified even for doing so once. Avi ha'Ezri holds that even murder disqualifies only if he does so repeatedly! The Pesul for Tum'ah is mid'Rabanan. It is not clear if we must be so stringent for doing so once.

Rema: Some say that if he has a daughter who became an apostate or was Mezanah, we are not obligated to sanctify him - "Ki Aviha Hi Mechaleles."

Mordechai (Sanhedrin 717): If a Kohen's daughter became an apostate, or was just Mezanah, it is optional (but no Mitzvah) for him to bless first.

Question (Magen Avraham 62): R. Shimon (Sanhedrin 50a) proved that burning is the most severe death, for a Bas Kohen who was Mezanah profanes her father, and is burned. If also idolatry profanes her father, there is no source to say that Zenus is more severe than idolatry)! The Mordechai said that apostasy profanes her father, for we assume that she was Mezanah. This is why he says so only about a daughter, and not about a son. Even though she left his Reshus when she became an apostate, he should have taught her beforehand.

Mishnah Berurah (153): He does not have precedence to begin and bless first, but he can Duchan. Even if he himself was Mezanah, this would not disqualify him from Birkas Kohanim!

Mishnah Berurah (154): If she was single, he is not disqualified. Nowadays, the custom is not to disqualify due to Zenus of apostasy of his children.