Gemara

Mishnah: On Monday, Thursday, and Minchah of Shabbos, exactly three read from the Torah. On Rosh Chodesh and Chol ha'Moed exactly four read. On Yom Tov five read; on Yom Kipur six read; on Shabbos seven read. We may not diminish from this, but Aliyos may be added.

22b - Beraisa: The general rule is:

Any day on which people may work, such as fasts and Tish'ah b'Av, three people read;

Days when people do not work, such as Rosh Chodesh and Chol ha'Moed, four read.

The general rule is, any day which has a greater Kedushah has more Aliyos;

There is Musaf on Rosh Chodesh and Chol ha'Mo'ed, therefore four read. Melachah is forbidden on Yom Tov, therefore five read...

Rishonim

The Rif (11b and 12b) and Rosh (3:1, 3:3, 3:4) bring our Mishnah and the Beraisa.

Ran (11b DH ha'Korei): Monday and Thursday are workdays, so we do not add Aliyos, for this inconveniences the Tzibur. At Minchah on Shabbos we do not add because the custom was to expound the entire day, and Minchah was shortly before dark.

Ran (12b DH Masnisin): On Rosh Chodesh we do not add Aliyos because Melachah is permitted. The same applies to Chol ha'Moed, for Devar ha'Eved (something that must be done to avoid a loss) is permitted. Rashi explains that the Heter to add Aliyos applies to Yom Tov and Yom Kipur as well as to Shabbos, for Melachah is forbidden on all of these. Others explain that the Isur to have less Aliyos applies also to Yom Tov and Yom Kipur, and the Heter to add is only on Shabbos, for we should not equate Yom Tov or Yom Kipur to Shabbos.

Note: We may not add even one Aliyah on Yom Tov for this would equate it to Yom Kipur.

Ran (DH b'Sish'ah): The Beraisa calls Rosh Chodesh a day when people do not work for this was the custom. Except for Devar ha'Eved, work is forbidden on Chol ha'Moed.

Rambam (Hilchos Tefilah 12:16): At Shacharis of Shabbos, Yom Kipur and Yom Tov, the number of Aliyos is seven, six and five, respectively; Aliyos may be added. On Rosh Chodesh and Chol ha'Mo'ed four read; at Minchah of Shabbos, Yom Kipur and Ta'aniyos, and at Shacharis of Mondays and Thursdays and Chanukah, Purim, and Ta'aniyos, three read; we do not add Aliyos.

Kesef Mishneh: The Rambam holds like Rashi (who permits adding even on Yom Tov and Yom Kipur). At Minchah of Yom Kipur we may not add for it is no better than (Minchah of) Shabbos.

Question (Kesef Mishneh): Ta'anis 12b mentions Kri'as ha'Torah at Minchah of Ta'aniyos, but not at Shacharis. What is the Rambam's source that we read at Shacharis?

Rosh (3:4): Ta'anis Tzibur and Tish'ah b'Av are considered days on which people work, since Melachah is Asur only in the middle and latter sets of Ta'anisim for lack of rain. Melachah on Tish'ah b'Av is forbidden only where this is the custom. Rosh Chodesh is called a day when people do not work, for the custom is that women do not work then (Yerushalmi). Targum Yonason (Shmuel 1:20:19) connotes that Rosh Chodesh is not considered a workday (even for men).

Mordechai (806): A case occurred, on Thursday there were three Chasanim (on the day of their Chupah) in the Beis ha'Keneses. One was a Kohen and two were Yisraelim. The Ritzva permitted to add a fourth Aliyah. Presumably, this is because laws of Mo'ed apply to a Chasan, e.g. he may not show signs of mourning in public (Kesuvos 4a), and four read during the Mo'ed. We would not read four if there was only one Chasan, for it is not Mo'ed for the others. All four who read were Chasanim, i.e. the Kohen read twice.

Poskim

Shulchan Aruch (OC 135:1): On Monday, Thursday, and Minchah of Shabbos, exactly three read from the Torah; we do not add Aliyos.

Rema: If there are two Chasanim Yisraelim it is permitted to read a fourth, since for them it is like Yom Tov on which it is permitted to add. It appears that the same applies when there are two Ba'alei Bris (the father, Sandek or Mohel), for it is like Yom Tov for them.

Rebuttal (Taz 1): We cannot add Aliyos because it deducts from the work of Yisraelim, and it is not Mo'ed for them! If it is like Yom Tov, they should also read a Haftorah!

Note: Our text of the Mordechai says that it is like Mo'ed, on which four read. Surely, he refers to Chol ha'Mo'ed.

Mishnah Berurah (3): Our custom is not to add for Chasanim, and certainly not for Ba'alei Bris.

Kaf ha'Chayim (2): There are Kabalistic reasons for exactly three Aliyos on Mondays and Thursdays.

Shulchan Aruch (282:1): On Shabbos seven read; we may add to this.

Magen Avraham (1): All Poskim permit to add even nowadays when every Olah blesses, for also in the days of the Gemara every Olah blessed. However, Tashbatz (2:70 b'Hagahah) says that we should not add because it is an unnecessary Berachah. It is proper to follow this, unless there is a Chasanah or Bris.

Mishnah Berurah (5): One should not add many Aliyos for this delays the Tzibur. If people will be resentful that they did not receive Aliyos, we may add many.

Rema: Similarly, on Yom Tov we may add. Some say that we may not add on Yom Tov. This is our custom, except for Simchas Torah.

Source (Gra DH v'Yesh): This opinion learns from (22b) 'any day which has a greater Kedushah has more Aliyos'. This is also the reason we may not add on days that three or four read (it is not due to Bitul Melachah).

Taz (1): Levush says that some places do not add Aliyos on Shabbos that is Yom Tov. This is folly. It is no worse than a standard Shabbos just because it is more Kodesh!

Magen Avraham (2): When Yom Kipur is on Shabbos some say that we should not add, for the seven Aliyos each end with matters of Kaparah, and adding Aliyos would necessitate ending in different places.

Mishnah Berurah (5): When two Parashiyos are read on Shabbos, half of the Aliyos should be from each Parashah, i.e. the fourth reads at least three verses from each Parashah. It is no problem that all the additional Aliyos are in the second Parashah, for they are not obligatory.

Kaf ha'Chayim (16): Some say that half the total (including the additional) number of Aliyos should be from each Parashah. Nevertheless, one should not change the custom in a place where only three and a half are from the first Parashah.

Mishnah Berurah (7): On Simchas Torah we rely on the majority of Poskim who permit adding on Yom Tov, in order to enable everyone to have an Aliyah at the time of the Siyum of the Torah.

Shulchan Aruch (2): Many Olim may read the same verses. Each blesses, and it is not a problem.

Source (Beis Yosef DH Kosav): Rivash (Teshuvah 84) permits, just like we reread the same verses on Chanukah and Chol ha'Mo'ed Sukos.

Kaf ha'Chayim (19): If possible, one with Yir'as Shamayim should avoid receiving an Aliyah if he will not read new verses.

Mishnah Berurah (9): One does not count towards the required number of Aliyos unless he reads at least two new verses.

Note: The Mishnah Berurah does not discuss when it was enacted to reread the same verses, e.g. Chanukah and Chol ha'Mo'ed Sukos.

Rema: Some forbid, and this is our custom, except for Simchas Torah.

Source: The Mordechai (831) says that we should not repeat verses needlessly. Rav permitted with difficulty to repeat one verse in the Kri'ah for Rosh Chodesh.

Source (Gra DH v'Yesh): Regarding Rosh Chodesh (22a) it says that we may repeat verses only when there is no alternative. The first opinion explains that this applies to obligatory Aliyos, not to additional Aliyos.