Gemara

Question: What is the source that a Kohen reads the Torah first?

Answer (R. Chiya bar Aba): "v'Kidashto (you will sanctify him)" - for every matter of Kedushah.

(Abaye): Why does the Mishnah say that it is for Darchei Shalom? We expound this from the Torah! We understand this from what Rabah said (the enactment requires that a Kohen read first. The Torah requires only asking him to read first.)

(Beraisa): The one who blessed on the bread, he takes first from the bowl. He may honor his Rebbi or one greater than himself by letting them take first.

(Rabah): This applies only to a meal. Regarding Kri'as ha'Torah, a Kohen may not honor another by letting him read first, lest this lead to quarrels.

(Rav Masnah): Rabah's law applies only on Shabbos or Yom Tov, when many people are in the Beis ha'Keneses, but not on Mondays or Thursdays.

Question: Rav Huna (who was not a Kohen) read first on Shabbos and Yom Tov!

Answer: He is different. (No one will quarrel if he reads first. Even R. Ami and R. Asi, the greatest Kohanim of Eretz Yisrael, were submissive to him!

60a (R. Yitzchak Nafcha): After the Kohen and Levi, we call Chachamim appointed over the Tzibur; Chachamim qualified to be appointed; children of the appointed Chachamim; the heads of the Tzibur; and everyone else.

Megilah 28a: One merit to which R. Freida attributed his long life was that he never blessed (at a meal) before a Kohen.

Question: R. Yochanan taught that if a Chacham allows an ignoramus Kohen, even a Kohen Gadol, to bless before him, he is worthy of death! We read "Kol Mesan'ai Ohavu Maves" like 'Masni'ai' (those who make people hate Torah).

Answer: R. Freida never blessed before a Kohen Chacham.

Rishonim

The Rif brings the teachings of R. Chiya bar Aba and R. Yitzchak Nafcha.

Rambam (Hilchos Tefilah 12:18): At every Kri'as ha'Torah, a Kohen reads first. The prevalent custom is that an Am ha'Aretz Kohen reads before a great Chacham Yisrael.

Perush ha'Mishnayos: There is no since in the Torah or Gemara for this. Where did this ruinous custom develop? The Torah gives precedence only when they are equal in Chachmah!

Beis Yosef (OC 135 DH v'Nir'eh): We cannot say that the custom developed through a mistake. The enactment for Shalom was even to call an Am ha'Aretz before a Chacham. A Chacham Mamzer comes before a Kohen Am ha'Aretz (Horayos 13a) for everything except Kri'as ha'Torah.

Rosh (5:20): A Kohen may not honor his Rebbi or a bigger Chacham than himself to read first on Shabbos or Yom Tov, when many are in the Beis ha'Keneses.

Tosfos (59b DH d'Afilu): Rav Huna read first only because the greatest Kohanim were submissive to him. What was the Chidush that Rav Freida never blessed before a Kohen? We must say that the greatest Kohanim were submissive to him. Alternatively, he did not read first even on Monday or Thursday.

Mordechai (401): Mid'Oraisa, Yisre'elim have a Mitzvah to call a Kohen to read first. A Kohen may pardon his honor. For Darchei Shalom he must read first, lest there be fights, e.g. on Shabbos. Nowadays we do not work, and many are in the Beis ha'Keneses even on Mondays and Thursdays, so the same applies then.

Poskim

Shulchan Aruch (OC 135:3): A Kohen reads first in the Torah.

Shulchan Aruch (4): The prevalent custom is that a Kohen Am ha'Aretz reads before the biggest Chacham in Yisrael, as long as the Kohen knows how to read.

Beis Yosef (DH Aval...): The Tur brings Rav Amram Gaon and R. Natrunai Ga'on, who forbid even the Nasi of Yisrael to read before a Kohen, even an Am ha'Aretz. Why does Maharik say that all Poskim allow the Gadol ha'Dor to read first? Perhaps this is when the Kohen consents. The Kol Bo says that an Am ha'Aretz Kohen reads before a Chacham Yisrael, but the Kohen may honor a great person to whom everyone is submissive.

Rebuttal (Taz 3): If Rav Amram and Rav Natrunai permitted when the Kohen consents, they would have said so! Also, the Rosh discusses when the Kohen consents, and the Tur says that the Rosh is unlike these Ge'onim! The Ge'onim learn from 60a. After the Kohen and Levi, we call Chachamim appointed over the Tzibur ... This shows that we do not call the leaders of the Tzibur before a Kohen. This is unlike Rav Huna.

Beis Yosef (DH v'Chosav... citing Maharik 9:1): In a certain congregation, the one who pledged the most money would read first on Parshas Bereishis. If he was a Yisrael, the Kohen would leave the Beis ha'Keneses. Once, a Kohen refused to leave. They forcibly kept him outside the Beis ha'Keneses, in order to preserve their custom. Likewise, if the Kohen is not fasting on BaHaB (optional fasts after Nisan and Tishrei), he leaves and a Yisrael who is fasting is called first.

Rebuttal (Beis Yosef, ibid.): Why must the Kohen leave?! He may stay and pardon his honor! The Kohanim did not leave when Rav Huna read. If they did, anyone could read first! If the Kohanim left only for him, the Gemara would have said so. A Kohen must leave only if he was asked to give Birkas Kohanim and will not do so, lest he transgress Mitzvos Aseh. He does not transgress anything by not reading first. V'Kidashto is a Mitzvah on Yisre'elim to honor him! People will not think that he is a Chalal. That will realize that he pardoned his honor for the custom of the Beis ha'Keneses or for a Chacham, or he is not fasting.

Chasam Sofer (OC 24 DH Aval): One may not make a new custom (to allow a Yisrael to read first) without consent of the Kohanim, due to v'Kidashto.

Kaf ha'Chayim (14): Some places normally call the Rav first on Shavu'os, even if he is a Yisrael. Ashkenazim honor visitors by calling them to the Torah. If there are many, the Kohen pardons his honor. Since he pardons only for guests, it will not lead to fights.

Mishnah Berurah (9): This applies whenever we read the Torah, not only on Shabbos and Yom Tov.

Mishnah Berurah (11): Many Rishonim say that there was never an enactment to put a Kohen Am ha'Aretz before a Chacham. However, the custom is like those who say that this was enacted, for Darchei Shalom.

Rema: It suffices that he can read word for word with the Ba'al Korei

Source (Darchei Moshe 4): The Avudraham says so. The Maharil says that the custom is to call a Kohen even if he cannot read word for word with the Ba'al Korei. This is wrong.

Mishnah Berurah (14): If another Kohen can read by himself, we call him.

Shulchan Aruch: If the Kohen cannot read, he cannot bless!

Mishnah Berurah (15): Nowadays we are not particular about this. We assume that everyone can read (if not by reading the words, in any case) orally with the Shali'ach Tzibur.

Magen Avraham (282:6): Some say that if the only Kohen is a minor, we call him to read first from the Torah. I say that v'Kidashto does not apply to a minor. "Lechem Elokecha Hu Makriv: does not apply to him.