Gemara

(R. Yehudah bar Nachmani): It says "Kesov Lecha Es ha'Devarim ha'Eleh", and "Ki Al Pi ha'Devarim ha'Eleh";

This forbids learning Oral Torah that was written, and learning written Torah (Torah, Nevi'im and Kesuvim) by heart.

Rishonim

Rif and Rosh (Megilah 14a and Megilah 3:10): A Tosefta teaches that if there is only one who knows to read, he stands, reads, and sits, stands, reads, and sits, even seven times.

Rosh (10): This proves that one who does not know how to read may not have an Aliyah, for it is a Berachah l'Vatalah. If he were permitted to have an Aliyah, this would be better than one person having seven Aliyos!

Rambam (Hilchos Tefilah 12:8): One may not read even one word (of Kri'as ha'Torah) not from the Sefer.

Rosh (Teshuvah 3:12): It is improper for an Am ha'Aretz (ignoramus) to be called to the Torah. Since he blesses, he must read, lest his Berachah be l'Vatalah. However, if he reads with the Ba'al Korei, and he understands to join the letters and read with him, this is called reading. However, he cannot bless on behalf of the Ba'al Korei's reading if he will not read at all.

Question (Tosfos 14b DH Devarim): Why may we say Pesukei d'Zimra by heart?

Answer #1 (Tosfos ibid.) The Isur is only when one is Motzi others (e.g. Kri'as ha'Torah).

Source (Tosfos Yeshanim Yoma 70a DH ube'Asor): Ta'anis 28a teaches that the Isur is when one is Motzi others.

Answer #2 (R. Tam, brought in Beis Yosef OC 49 DH u'Mah): Something that everyone knows well is permitted. It is even permitted to write Tefilin by heart because people know the Parashiyos.

Answer #3 (Tosfos Yeshanim ibid.): There is no Isur. It is merely better to read written Torah than to say it by heart.

Tosfos (Sotah 39a DH Kivan): At the time of Kri'as ha'Torah, Rav Sheshes turned away and learned Halachos. Since he was blind, he was exempt from Kri'as ha'Torah, since one may not say written Torah by heart.

Poskim

Shulchan Aruch (OC 139:1): In a place where the Oleh himself reads aloud, if he did not prepare (read) the Parshah two or three times by himself, he may not have an Aliyah.

Beis Yosef (DH ha'Korei): We learn from the case of R. Akiva (Medrash Tanchuma Yisro 15. He was called to the Torah and refused to read, because he had not prepared the Parshah. The Medrash Rabah (Ki Sisa 40b) says that R. Yochanan was called to read in front of R. Akiva, and he refused for this reason.) This was in their days, when the Oleh himself read aloud. Our custom is that he does not read aloud, therefore even if he did not prepare the Parshah, there is no problem. The Ba'al Korei must prepare the Parshah.

Gra: The Shulchan Aruch rules like Medrash Tanchuma. Medrash Rabah says that one should prepare it four times

Mishnah Berurah (1): Even though if one is called to the Torah and he does not read, this shortens his life, here he should not read, for one must prepare beforehand. Chachamim found an Asmachta for this - Az Ra'ah va'Ysaprah.

Mishnah Berurah (2): If no one prepared, rather than omit Kri'as ha'Torah, we let read one who knows the (musical) notes, even though he did not prepare. We read quietly before him from a Chumash or Siddur.

Rema: In a place where the Ba'al Korei reads, he must prepare the Parshah.

Bach (1): The ideal Mitzvah is that the Oleh prepares the Parshah even though the Ba'al Korei reads it aloud. Therefore, everyone should prepare the Parshah, lest he be called to read on Shabbos.

Mishnah Berurah (3): Learning Shnayim Mikra v'Echad Targum on Erev Shabbos counts towards preparation.

Kaf ha'Chayim (2): If he prepared two or three times, but still errs, he need not do more, unless he desires the ideal Mitzvah.

Shulchan Aruch (2): If one does not know how to read, we must protest and not allow him to have an Aliyah.

Beis Yosef (DH veha'Rav): From R. Akiva, we learn all the more so that one may not ascend to read from the Torah if he cannot read. We must protest that he not ascend.

Gra (DH Mi): The Avudaraham learns this from the previous Sa'if. He must hold that even where a Ba'al Korei reads for everyone, the Oleh must prepare the Kri'ah (so he will be able to read it), unlike the Beis Yosef. However, the Rosh and Rivash learned from the Tosefta that the Rif and Rosh brought.

Mishnah Berurah (4): The Mechaber connotes that even if he can read word for word with the Ba'al Korei, we may not call him to the Torah, since we can call someone else. However, the Tur (141) and Teshuvas ha'Rosh connote that one may be lenient about this, especially according to what (the Shulchan Aruch) wrote in Sa'if 3 in the name of Maharil

Mishnah Berurah (5): This is because one may not read even one word of the Torah not from the Kesav (writing). However, our custom is to be lenient about this; see Sa'if 3.

Shulchan Aruch (ibid.): If we need him, for he is the only Kohen or Levi, if when the Ba'al Korei says each word he can say it and read it from the Kesav, he may have an Aliyah. If not, he may not.

Beis Yosef (DH veha'Rav): Rav Sadya Gaon says that if we need him, for he is the only Kohen or Levi, if when the Ba'al Korei says each word he knows to say it, he may have an Aliyah. If not, he may not.

Magen Avraham (1): Mas'as Binyamin permits even if he cannot read from the Kesav.

Gra (DH v'Im): This is like regarding Bikurim (Mishnah 3:7). One who does not know how to recite the Parshah (of Viduy Bikurim, i.e. Devarim 26:3-10, we are Makreh, i.e. say the verses and he repeats them), and he is Yotzei the Mitzvah of reciting.

Shulchan Aruch (3): A blind person may not read, because one may not read even one word not from the Kesav.

Gra (DH Suma): Megilah 24a says that a blind person may say the Berachos of Shma (for the Tzibur) and translate the Torah. This implies that he may not read the Torah. See Tosfos in Sotah.

Rema: The Maharil says that nowadays a blind person may read, just like we are Makreh for an Am ha'Aretz.

Magen Avraham (4): Masas Binyamin proved that nowadays that the Ba'al Korei reads, one may call him. Mateh Moshe and the Levush agree. The Bach says that the custom is to call a blind person, but not an Am ha'Aretz. The custom is not to be particular about this, for presumably anyone can read with the Ba'al Korei.

Gra (DH u'Maharil): This is like the Tosefta (Megilah 2:2) and the Yerushalmi, which say that one may bless and another reads, like the Rema says in 692:1 (about Kri'as ha'Megilah). A blind person may not read for the Tzibur (since he does not see the words), but he may bless. I do not understand the Rosh in Megilah. In the days of the Gemara, only the first and last Olim blessed (and everyone else read for himself). If so, why should a blind person be Oleh?! This requires investigation.

Damesek Eliezer: The Rosh's proof is from the fact that the one who knows to read receives all seven Aliyos. If one can bless and another will read, others could receive the first and last Aliyos, and the one who reads need not receive more than the middle five Aliyos!

Mishnah Berurah (12): Since nowadays the Ba'al Korei reads, and he reads from the Kesav, we are not concerned whether the Oleh reads from the Kesav, for Shome'a k'Oneh (one who hears is like one who says).

Mishnah Berurah (13): We are lenient about an Am ha'Aretz even if he cannot read with the Ba'al Korei, so the same applies to a blind person. The custom is to be lenient like Maharil, but one should not call him for Parshas Parah or Zachor.

Kaf ha'Chayim (15): In many places the custom is to be lenient like the Rema. Where there is no custom, one must be stringent like the Mechaber, for we accepted his rulings, and to avoid a Safek Berachah. Even where they are lenient, they should be lenient only if he is the only Kohen, or a Chasan, or he his son will be circumcised today.

Kaf ha'Chayim (16): Where they follow the Mechaber, if a Chasan is blind, he can ascend to the Bimah for the mere appearance of having an Aliyah, but the Ba'al Korei says the Berachos.

Shulchan Aruch (141:2): Two may not read. Either the Oleh reads and the Ba'al Korei is quiet, or the Ba'al Korei reads and the Oleh does not read aloud. In any case he must read with the Ba'al Korei, lest his Berachah be l'Vatalah. Rather, he reads softly, so he is not audible to himself.

Taz (3): The Beis Yosef learns from the Rosh. However, a Tosefta says that R. Meir read and allowed someone else to bless! The Yerushalmi questioned this, and answered that Shome'a k'Oneh. It seems that when we forbid a blind person to be Oleh, this is to read for the Tzibur, but he may bless and another reads. The Beis Yosef brought some who permit, and some who forbid. I proved that it is permitted.

Kaf ha'Chayim (16): Some Acharonim explained why Shome'a k'Oneh does not apply here. Also the Ari Zal says that the Oleh must read softly, (It is not enough to hear the Ba'al Korei.)

Bi'ur Halachah (DH l'Vatalah): Above, the Rema permits a blind person to have an Aliyah. Why didn't he say here that if he does not read, it is not a Berachah l'Vatalah? Perhaps he relied on what he said above. It is better to say that the Rema relies on the Maharil only for a blind person and Am ha'Aretz, for if we will never call them to the Torah, there is great shame and animosity. One who can read must read. The Darchei Moshe sided with those who forbid one to bless if he will not read, unlike Maharil! The Taz brought a proof from the Yerushalmi, which discusses Kri'as ha'Megilah. There is different, for the entire Tzibur is obligated to read and bless! Therefore, one may bless for all of them. The Tzibur may not bless for hearing Kri'as ha'Torah! Perhaps the Taz holds that the Ba'al Korei blesses only for the Oleh, and Shome'a k'Oneh applies. The Rosh forbids based on his opinion that the enactment (of Kri'as ha'Torah) was for the Tzibur. The Chayei Adam and Sha'arei Efrayim hold like the Shulchan Aruch here.

Gra (5): Tosfos (Megilah 21b DH Tana) says that the Ba'al Korei reads to avoid shaming one who does not know how to read. The Rosh (Megilah 3:1) says that it is because some do not know that one may not read by heart, and the Tzibur would not be Yotzei through them.

Mishnah Berurah (8): Some do not know to read the vowels and notes properly. If we would protest against them, this would lead to strife.

Kaf ha'Chayim (15): Even one who knows how to read should not read, lest he embarrass those who cannot.

Gra (6): Also in the days of the Gemara, the Ba'al Korei used to help the Oleh read (Rashi Shabbos 12b DH Roshei).

Mishnah Berurah (7): When the Oleh reads aloud, if he errs the Ba'al Korei must quietly help correct him.

Rema: Even if he is audible to himself, there is no problem. This is not more stringent than Tefilah.