Gemara

Talmud Torah is forbidden to an Avel, for Hash-m told Yechezkeil "Dom".

21a - Beraisa: An Avel may not read Torah, Nevi'im, Kesuvim, or learn Mishnah, Medrash, Halachos or Agados. If many need him he does not refrain. When R. Yosi's son died, R. Yosi entered the Beis Medrash and taught the entire day.

When Rabah bar bar Chanah became an Avel, he thought that that he would not go to teach. R. Chanina told him that if many need the Avel, he does not refrain. Rabah thought that that he would use a Meturgeman (one who translates and broadcasts the lecture to Talmidim). Rav told him that a Beraisa forbids this.

Question: How can he teach without a Meturgeman?

Answer (Beraisa): When R. Yehudah's son died, R. Yehudah entered the Beis Medrash. R. Chanina ben Akavya sat next to him. R. Yehudah whispered to R. Chanina, and R. Chanina told the Meturgeman, and the Meturgeman broadcasted it.

Ta'anis 30a - Beraisa: All Mitzvos that apply to an Avel apply on Tish'ah b'Av. One may not read Torah, Nevi'im, Kesuvim, or learn Mishnah, Talmud, Medrash, Halachos or Agados. One may read or learn something he is not familiar with, Iyov, Kinos and dire parts of Yirmeyah. Children are (most texts - not) Batel from learning. This is because "Pekudei Hash-m Yesharim Mesamchei Lev";

R. Yehudah forbids learning something one is not familiar with; only Iyov, Kinos and dire parts of Yirmeyah are permitted. Children are Batel from learning.

Rishonim

The Rif and Rosh (3:37) bring the teachings in Moed Katan.

Question: Why is an Avel exempt from Talmud Torah? He is obligated to keep all the Mitzvos!

Answer (Nimukei Yosef 13a DH v'Asur): Rashi answers that he is exempt from Mitzvos that give Simchah. He is forbidden to have relations for the same reason, even though it is a Mitzvah in the proper time.

Rif (Ta'anis 10a): The Halachah follows R. Yehudah, who forbids learning something one is not familiar with. Only Iyov, Kinos and dire parts of Yirmeyah are permitted. Children are Batel from learning.

Rosh (ibid.): On Tish'ah b'Av, one may read or learn something he is not familiar with, and Iyov and dire parts of Yirmeyah. This is our custom; also Hilchos Aveilus including the relevant part of Moed Katan is permitted. However, Evel Rabsi (Maseches Semachos 10) says that when a Chacham is an Avel, Talmidim learn Hilchos Aveilus in front of him, and if they err he corrects them solemnly. He may not ask questions. We can say that he may learn by himself, but in front of Talmidim is too public. This is when the Rabim do not need him. If the Rabim need him, he may teach even other Halachos.

Rambam (Hilchos Evel 5:16): An Avel may not read Torah, Nevi'im, Kesuvim, or learn Mishnah, Medrash, Halachos or Agados.

Hagahos Maimoniyos (40): The Yerushalmi permits an Avel to learn something he is not accustomed to, for this is (very taxing and) not Simchah. Avi ha'Ezri says that this supports the custom to allow an Avel to learn Hilchos Avelus. Also, the Gemara (Ta'anis 30a) says that everything that applies to an Avel applies on Tish'ah b'Av, and on Tish'ah b'Av we permit Iyov and dire parts of Yirmeyah, for these do not bring Simchah.

Radvaz: There is an argument about Iyov, Kinos and dire parts of Yirmeyah. I permit, for these will prevent him from being Batel without causing him to forget his mourning or to be jovial.

Rambam (ibid.): If many need him he is permitted, as long as he does not (directly) use a Meturgeman. Rather, he whispers to someone who then tells the Meturgeman.

Nimukei Yosef (DH Rabim): This refers to when he is needed to rule in Halachah and there is no one else who can, or if he is the Nasi and he expounds to the Rabim.

Poskim

Shulchan Aruch (384:1): All seven days an Avel may not read Torah, Nevi'im, Kesuvim, or learn Mishnah, Gemara, Halachos or Agados. If many need to learn from him he is permitted, as long as he does not set up a Meturgeman. Rather, he whispers to someone who then tells the Meturgeman, and the Meturgeman broadcasts it.

Orchos Chayim (brought in Bedek ha'Bayis and Taz 1): My Rebbi'im would teach after three days. The same Heter applies to one who teaches children, for they need him, and their Torah is greater than that of adults. Also, it is Devar ha'Eved if they do not learn.

Rema: Alternatively, he may expound out loud to everyone. He may give rulings in Isur v'Heter if there is no one else, but he may not teach Halachah to his Talmidim. This is the custom, even though some are lenient.

Shulchan Aruch (2): If the only Kohen in the Beis ha'Keneses is an Avel, he may not receive an Aliyah to read from the Torah.

Source (Beis Yosef DH u'Mah she'Chosav b'Shem): The Ritz Gei'us says that an Avel may read Torah only if the Rabim need him. Therefore, a Kohen Avel may not be called to the Torah, even if there is no other Kohen. A Kohen reads first only to avoid strife. A Yisrael reads first when there is no Kohen, or if the Yisrael is greater in Torah. The Rambam says that the Avel does not even leave his house to go to the Beis ha'Keneses.

Bach (DH v'Chosav): The Ritz Gei'us teaches about on Shabbos, or when there is a Sefer Torah in Beis ha'Evel. However, the Rema (400:1) permits him to have an Aliyah on Shabbos if he is the only Kohen.

Shach (3): Here the Shulchan Aruch teaches what is forbidden to an Avel. Siman 400 teaches how we avoid public Aveilus on Shabbos.

Shulchan Aruch (4): He may read Iyov, Kinos, dire parts of Yirmeyah, and Hilchos Aveilus.

Beis Yosef (DH Yesh Omrim): The Ramban and Tosfos bring opinions that since the Heter for these matters was taught only regarding Tish'ah b'Av, the Heter does not apply to an Avel. Similarly, Aveilus is more stringent regarding turning over the bed, covering the head and Melachah.

Gra (OC 554:2): He may learn these with Perushim.

Shulchan Aruch (ibid.): He may learn these by himself, but he may not learn with others. If others are learning Hilchos Aveilus and they erred, he may correct them in a serious tone. He may not ask. If many need to learn from him he is permitted even other subjects, like we said.