Gemara

(Mishnah): Outside the Mikdash, the Kohanim lift their hands to shoulder height.

(Beraisa): "So you will bless" - with lifted hands.

(R. Nasan): "He (Aharon) and his children, for all days" equates Aharon's children to Aharon. Just like Aharon raised his hands, also his children raise their hands.

Chagigah 16a (R. Yehudah b'Rebbi Nachmani): If one looks at the Kohanim, his eyes weaken.

This is when the Mikdash stood. The Kohanim stood on the Duchan and blessed Yisrael with Shem ha'Mefurash.

Megilah 24b (Mishnah): A Kohen who has Mumim on his hands may not Duchan;

R. Yehudah forbids even if his hands are dyed, because people look at him.

(Beraisa): Mumim on hands, feet and face disqualify a Kohen from Berachah.

Rishonim

Rif and Rosh (Megilah 15b and 3:18): Birkas Kohanim must be said with lifted hands.

Rambam (Hilchos Tefilah 14:7): When Kohanim bless the people, they may not look at the people, lest this distract them. Rather, their eyes must be towards the ground, like one who is praying. The people may not look at the Kohanim, lest this distract them. Rather, everyone intends to hear the Berachos and points his face towards the Kohanim, but does not look at their faces.

R. Mano'ach: Also the Tzibur faces down, like during Tefilah. The Yerushalmi says that the Kohanim may not look at the people, due to Hesech ha'Da'as.

Kesef Mishneh: In every place one may not look at the Kohanim's faces, lest this distract him (see Tosfos below). All the more so, we are concerned lest Kohanim be distracted!

Rambam (11): A tradition from Sinai obligates blessing with lifted hands.

Rambam (15:2): If a Kohen has Mumim on his hands, feet or face, he may not give Birkas Kohanim, for people will stare at him.

Tosfos (Chagigah 16a DH ba'Kohanim): Rashi says that we are concerned lest people stare at his hands, for this weakens the eyes. This is only in the Mikdash, yet we forbid one with Mumim on his hands even outside the Mikdash! Rather, the Yerushalmi explains that we are concerned lest they stare and be distracted.

Or Zaru'a (2, Sof Siman 411): R. Yosi said that he can look at the Kohanim and not be distracted. Such people may look at the Kohanim.

Poskim

Shulchan Aruch (OC 128:14): Birkas Kohanim is said only with lifted hands.

Mishnah Berurah (50): Almost all the Poskim say that this is Me'akev. If it was not fulfilled, even b'Di'eved they were not Yotzei. If a Kohen cannot bless properly, he should leave the Beis ha'Keneses before Birkas Retze.

Mishnah Berurah (52): If a Kohen can lift his hands only if they are supported, he may not bless. However, it suffices to lift them while saying each word of the Berachos. He may lower them in between words.

Shulchan Aruch (23): When Kohanim bless the people, they may not look, lest this distract them. Rather, their eyes must be towards the ground, like one who is praying. The people intend to hear the Berachos and point their faces towards the Kohanim, but do not look at their faces.

Magen Avraham (35): All the more so, they may not look elsewhere. To be Mistakel, i.e. to gaze, is forbidden; a mere glance is permitted. The custom is not to look at all. Perhaps this is a commemoration of the Mikdash, in which even a glance is forbidden due to honor of the Shechinah. If one gazed (at their hands) in the Mikdash, his eyes would weaken.

Kaf ha'Chayim (140): The Zohar says that even nowadays one may not even glance elsewhere due to honor of the Shechinah.

Mishnah Berurah (88): The Kohanim face down like one who is praying, for they pray that Hash-m bless Yisrael.

Kaf ha'Chayim (139): When there are no Kohanim and the Shali'ach Tzibur says 'Elokeinu vEilokei Avoseinu...', people should face the Shali'ach Tzibur.

Rema: Also the Kohanim may not look at their own hands. Therefore, the custom is to lower the Talis over their faces and their hands are outside the Talis. In some places the custom is have their hands inside the Talis, so people will not look at their hands.

Beis Yosef (DH uvi'Sha'as): In Mitzrayim, they lower a Talis of a Mitzvah over their faces and hands. This is a good custom, lest they look at the people or people look at them and be distracted.

Gra (51): Megilah 24b supports the former custom. See Sa'if 31.

Note: The Mishnah disqualifies Mumim on the hands, but not on the face (like the Beraisa does), for it discusses a place where Kohanim cover their faces with a Talis. If the hands were inside the Talis, we would permit even Mumin on the hands! (Based on Dameshek Eliezer.)

Kaf ha'Chayim (142): In every place it is proper for people to cover their faces with a Talis or to close their eyes.

Shulchan Aruch (31): If the local custom is for Kohanim to lower a Talis over their faces, even if one has many Mumim on his face and hands, he may bless.

Beis Yosef (DH v'Nireh): We decree lest he bless where Kohanim do not lower a Talis over their faces, for letter of the law, even that is permitted. We forbid lest people be distracted. However, where others do not cover their faces, he may not, for this will cause people to look at him and be distracted, and all the more so if he is alone.

Magen Avraham (45): If it is not the custom, it is forbidden even if all the Kohanim will do so, for this will distract people.

Rema: This is only if his hands were inside the Talis. If they were outside, the Talis does not help for Mumim on his hands.

Question (Taz 28): The Rema should have taught that nowadays that also the Tzibur are careful to cover their faces themselves with Taleisim, it is permitted! This is better than Kohanim covering their faces!

Answer (Mishnah Berurah 115): If people know that a Kohen has a Mum, they will look. Also, single men do not have Taleisim.