Gemara

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

Sanhedrin 35b (Abaye) Suggestion: Avodah overrides Shabbos, but Misas Beis Din does not override Shabbos. All the more so, Misas Beis Din should not override Avodah!

Rejection: It says "me'Im Mizbechi Tikachenu Lamus"! (If a Kohen murdered, even if he is near the Mizbe'ach and about to do Avodah, we immediately take him to execute him.)

Yerushalmi (Gitin 5:9 23b): Do not say 'Ploni the Kohen transgresses Arayos and murder, and he blesses us?!' Hash-m says 'who blesses you? I am the One who blesses you - "va'Ani Avarachem"!'

Rishonim

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

Hagahos Maimoniyos (1): Avi ha'Ezri disqualifies only if he is known to do so repeatedly. R. Simchah brings a proof from the Yerushalmi. Sefer ha'Mitzvos (20) wrote like the Rambam.

Mordechai (Megilah 818): Avi ha'Ezri disqualifies only if he is known to do so repeatedly, due to the Yerushalmi. It connotes that a Kohen who killed may Duchan. If a Kohen circumcised a baby and the baby died, R. Shimshon says that he can Duchan. The Gemara connotes that only one who killed b'Shogeg b'Mezid is disqualified, for it says Yedeichem Damim Male'u. Here is different, for he intended for a Mitzvah. Also, perhaps the baby was not born after a full term pregnancy (it was a Nefel)! Also, perhaps wind (or a spirit) confounded the baby (and caused its death).

Rambam (Hilchos Rotze'ach 6:3): If he was Shogeg close to Ones, i.e. it was wondrous that he died, for usually people do not die in such cases, he is exempt from Galus, and if the Go'el ha'Dam killed him, he is killed for this.

Rivash (94): If a Kohen killed someone, he forfeits the Kedushah. All the more so this applies nowadays, when Kohanim cannot prove their lineage.

Tosfos (Sanhedrin 35b DH she'Ne'emar): Why does the Gemara say that execution overrides Avodah? The Kohen is not qualified to serve! A murderer may not Duchan due to "Yedeichem Damim Male'u", all the more so he may not do Avodah! We must say that his Avodah is Kosher. It is a mere stringency that he does not Duchan. Alternatively, Birkas Kohanim is different, for his hands killed, and the prosecutor cannot become the advocate. The same verse that disqualifies from Birkas Kohanim says "uv'Prishchem Kapeichem A'alim Einai."

Ritva (Makos 13a DH Mai): Tosfos and my Rebbi (the Ramah) say that a Kohen who killed may not Duchan even if he repented, for it says Yedeichem Damim Male'u.

Tur (OC 128): Rashi and R. Gershom say that if he repented, he may Duchan. He brings a proof from the Yerushalmi. Maharam me'Rotenburg says that we do not tell him to ascend, but if he ascended, we do not protest. It seems that l'Chatchilah he should not bless, because Birkas Kohanim is equated to Avodah.

Poskim

Shulchan Aruch (OC 128:35): If a Kohen killed someone, even b'Shogeg, he may not give Birkas Kohanim, even if he repented.

Taz (32): "Yedeichem Damim Male'u... Gam Ki Sarbu Sefilah" connotes that Teshuvah does not help.

Be'er Heitev (59, citing Pri Chodosh 35): If he was forced to kill, he may Duchan.

Mishnah Berurah (128): Even though one must forfeit his life and not murder, if he killed he is not Chayav Misah for this, and he is not disqualified from Birkas Kohanim.

Be'er Heitev (59a citing Pri Chodosh 35): If he shoved a pregnant woman and she miscarried, he may Duchan.

Mishnah Berurah (129): This is because one is not Chayav Misah for killing a fetus.

Pri Chodosh (37): Teshuvah helps for a Mumar, but not for a murderer, for the prosecutor cannot become the advocate.

Kaf ha'Chayim (205): We prevent him from giving Birkas Kohanim only if there are witnesses that he killed.

Kaf ha'Chayim (208): If he was Shogeg close to Ones, he may Duchan even without Teshuvah. Eliyahu Rabah (63) learns from the Rambam in Hilchos Rotze'ach that if he was Shogeg close to Ones, he may Duchan, even without Teshuvah. Eshel Avraham (51) is stringent l'Chatchilah. Therefore, one should not be lenient until he repented.

Kaf ha'Chayim (209): When a Kohen is disqualified, he may not Duchanwhether or not there are other Kohanim who will Duchan.

Rema: Some say that if he repented, he may give Birkas Kohanim. We should be lenient for Ba'alei Teshuvah, and not close the door (to Teshuvah) in front of them. This is the custom.

Kaf ha'Chayim (210): Some say that the Mechaber holds that Teshuvah does not help even if he was Shogeg. Some say that this is only if he was Mezid. L'Chatchilah, he should leave the Beis ha'Keneses before Retzeh. If he ascended, he (surely, the text should say 'does not' - PF) descend. This is only if he repented, in a place where they conduct like the Rema.

Bi'ur Halachah (DH Afilu Asah): Below (Sa'if 37, regarding a Mumar), the Mechaber brought two opinions about whether or not Teshuvah helps. Here it is clear to him that we are stringent, for the prosecutor cannot become the advocate. Eliyahu Rabah and Mateh Yehudah say so. Olas Tamid and the Gra say that the Mechaber relied on what he wrote in Sa'if 37. The Rema is lenient, i.e. even if he transgressed b'Mezid. Several Acharonim bring this. The Pri Chodosh and Eliyahu Rabah are stringent if he was Mezid. The Ritva is stringent in every case. In any case, if he ascended, we do not make him descend.

Shulchan Aruch (36): If a Kohen circumcised a baby and the baby died, he may Duchan.

Source: The Mordechai and Hagahos Maimoniyos bring this from R. Shimshon, and R. Yerucham brings it from the Ramah (Beis Yosef DH v'Chasav).

Magen Avraham (52): Even when one killed an adult b'Shogeg and he did not die immediately, we can say that perhaps the wind confounded him (Gitin 70b). There is another claim if the victim quivered. (Perhaps he caused his own death, like the Gemara says there.)

Shulchan Aruch (ibid.): If people murmur that a Kohen kills, since it was not clarified, he may Duchan.

Beis Yosef (DH Kohen): The Rambam explains that the Yerushalmi discusses a Kohen who is not meticulous about Mitzvos, and everyone murmurs about him that he murdered. If we know that he killed, he may not Duchan.

Rebuttal (Shirei Korban Gitin 32b DH she'Lo): The Yerushalmi did not mention that we discuss murmuring. Also, 'Hashem says 'who blesses you?...'' connotes that the Kohen really killed. It is better to say that the Bavli argues with the Yerushalmi. (The Rambam rules like the Bavli.) Alternatively, it is a mere stringency that a murderer may not Duchan, like Tosfos says. Therefore, if the Kohen ascended, he does not descend. Alternatively, if they told him to bless, he blesses. R. Yochanan says that he should refrain. The Yerushalmi agrees. I think that this is the intent of Maharam me'Rotenburg that the Tur brings. We do not tell him to ascend, but if he ascended, we do not protest. The Tur connotes that we discuss one who repented. The Yerushalmi connotes unlike this. Based on what I wrote, this is not difficult.

Magen Avraham (53): If he himself knows that he killed, he may not Duchan.

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

Eshel Avraham (51, b'Sof): Tzedah l'Derech (on Rashi, Mishpatim 21:14) says that if a Kohen killed a Nochri, he does not Duchan. However, (I say that) if he killed someone whom we cast into a pit (to die), he may Duchan.

Ma'amar Mordechai (cited in Tzitz Eliezer 14:60): The Prishah disqualifies (only) one who killed a Yisrael. He learns from Yedeichem Damim Male'u, which discusses killing Yisre'elim. Why did the Prishah need to teach this? This is obvious! I answer that when we are not at war, we do not cause their death (YD 158), so one might have thought that this disqualifies. However, the Prishah brings from the Beis Yosef there that one need not cause their death, but one may! Rather, here we discuss even one who keeps his seven Mitzvos. One may not cause his death. Alternatively, even though one may not cause their death, one might have thought that this disqualifies from Birkas Kohanim because the prosecutor cannot become the advocate. The first answer is primary.

Tzitz Eliezer (ibid.): Tzedah l'Derech forbids a Kohen who killed a Nochri, and the Ma'amar Mordechai struggled to explain what is the Chidush that he may Duchan! These words of the Prishah are found in the Konigsburg edition of the Tur. They are not in our versions, due to censorship. This is why many who discussed this question did not mention his opinion. The Rema (CM 25:2) rules that if we find a Teshuvah of a Gaon that no one mentioned, and Acharonim disagree, we need not rule like the Acharonim, for had they seen the Teshuvah, perhaps they would have retracted. Perhaps this applies here, and especially because the Tzedah l'Derech was the Prishah's Talmid. All the more so, if a Kohen killed a Nochri in war, surely he may Duchan.

Kaf ha'Chayim (208): One who killed a Nochri is not disqualified.