Gemara

(Mishnah): The Kohen Gadol reads (Avodas Yom Kipur) in Acharei Mos and "Ach be'Asor" (in Emor). He rolls the Sefer Torah closed, puts it to his chest and says 'there is more here than what I read', and reads "Ube'Asor" in Bamidbar by heart.

70a - Question: Why does he read "Ube'Asor" by heart? He should roll the Sefer to there!

Answer (Rav Huna brei d'Rav Yehoshua): It is dishonorable to the Tzibur to roll a Sefer Torah b'Tzibur (unless it can be rolled to the next Parashah within the time that the translator translates the last verse, e.g. from Acharei Mos to "Ach be'Asor").

Question: Why don't they bring a second Sefer Torah and read from it?

Answer #1 (Rav Huna bar Yehudah): People might say that the brought a second Sefer because the first is Pasul.

Answer #2 (Reish Lakish): This would case an unnecessary Berachah (on reading from the second Sefer).

Gitin 60b - Question: It says "Kesov Lecha Es ha'Devarim ha'Eleh", and it says "Al Pi ha'Devarim ha'Eleh" (they should be oral)!

Answer (R. Yehudah bar Nachmani): You may not say Torah shebi'Chsav by heart, and you may not recite Torah she'Ba'al Peh by reading it (Rashi's text - you may not write Torah she'Ba'al Peh).

Ta'anis 28a - Beraisa: At Shacharis and Musaf, Ma'amados read the Torah in the normal way. At Minchah, an individual reads it by heart;

R. Yosi: An individual may not read Divrei Torah by heart b'Tzibur! Rather, everyone reads it by heart, the way we read Shma.

Rishonim

Rambam (Hilchos Tefilah 12:8): It is forbidden to read even one word (of Kri'as ha'Torah) not from the Sefer.

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

Answer #1 (Tosfos ibid.) The Isur does not apply to Nevi'im and Kesuvim, only to Chumash.

Objection: We say Shiras ha'Yam and Kri'as Shma by heart!

Answer #2 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 #3 (R. Tam, brought in Beis Yosef OC 49 DH u'Mah): Something which everyone knows well is permitted. It is even permitted to write Tefilin by heart because people know the Parashiyos.

Answer #4 (Tosfos Yeshanim ibid.): There is no Isur. It is merely better to read Torah shebi'Chsav than to say it by heart.

Question: The Mishnah (Berachos 9b) says that one who says Shma after the time does not lose, like one who reads Torah. This is obvious!

Answer (R. Yonah DH Lo Hifsid): It teaches that since it is permitted to say Shma by heart for the Chiyuv, it is always permitted. Likewise, we may say Parashiyos of Korbanos by heart, because one must say them every day. (Ta'anis 27b says that when there is no Beis ha'Mikdash, Yisrael will read the Parashiyos of Korbanos, and Hash-m considers it as if they built the Mikdash and offered Korbanos, and He pardons all their sins.) This permits the Kohen Gadol to read the Korbanos of Yom Kipur by heart, even though that reading is not obligatory. One may say Birkas Kohanim by heart, since it is obligatory when Kohanim Duchan.

Tosfos (Bava Kama 3b DH kede'Metargem): Some say that Rav Yosef often gives the Targum of verses because he was blind, and it was forbidden for him to say Torah shebi'Chsav by heart. This is wrong. When there is no alternative it is permitted because of "Es La'asos la'Shem". A blind person is the ultimate "Es La'asos" (otherwise, he could never learn say Torah shebi'Chsav).

Note: Perhaps Tosfos refers to the Heter to write a Sefer just of Parashiyos of the Haftorah because of "Es La'asos" (Gitin 60a). Since it is not a full Sefer, it is considered by heart. Alternatively, he refers to the Heter (ibid.) to write Sefer Agadata. The Isur to write Torah she'Ba'al Peh is learned from the same verse that forbids to say Torah shebi'Chsav by heart.

Poskim

Shulchan Aruch (OC 49:1): Even though it is forbidden to say Torah shebi'Chsav by heart, something which everyone knows well is permitted. Some examples are Kri'as Shma, Birkas Kohanim, and Parashas ha'Tamid.

Magen Avraham (1): If an individual knows a Parashah by heart but not everyone does, it is forbidden.

Kaf ha'Chayim (1): The Bach and others permit in such a case. One may rely on them to avoid Bitul Torah if there is no alternative.

Magen Avraham (1): One may write Tefilin by heart because all scribes know the Parashiyos. The Bach rules like the Tur, that the Isur is when one is Motzi others. This is why the custom is that everyone says "Mi Chamocha" and "Hash-m Yimloch" by themselves. The Shali'ach Tzibur used to pray by heart, and he was unable to be Motzi people. There are many opinions, therefore one should not say anything by heart except for what the Shulchan Aruch permits. It is better not to read the Parashah by heart with the Shali'ach Tzibur, but we do not protest against one who does. Some say that it is forbidden only when one reads Shnayim Mikra v'Echad Targum.

Beis Yosef (DH u'Mah): It is forbidden to say Torah shebi'Chsav by heart because there are many things which are expounded from the written text, such as extra letters and words read unlike they are written. The Kohen Gadol read by heart on Yom Kipur because we do not roll the Torah b'Tzibur, and to bring another Sefer Torah would disgrace the first one.

Beis Yosef (DH veha'Kolbo): The Kolbo says that we rely on "Es La'asos" to read Parashas ha'Tamid and Birkas Kohanim by heart, for not everyone can bring a Sefer to read them every morning. We say Tehilim 20 ("Ya'ancha Hash-m b'Yom Tzarah") quietly (except for the first verse) and also Kri'as Shma. We need not be concerned for two or three verses that everyone knows. Some say that the Isur is only if one reads with cantillation. The Yerushalmi says that the Isur is only when reading from a Sefer Torah, lest people say that the Sefer is lacking. We are not concerned during prayer.

Mishnah Berurah (3): If one expounds b'Rabim and it would be dishonorable to the Tzibur to look up all the verses to read them verses from a Sefer, he may be lenient.

Mishnah Berurah (6): Everyone is fluent in Pesukei d'Zimra. Chavos Ya'ir permits all of Tehilim. Since it is to arouse mercy, it is like prayer. One may rely on this, for the Gra and other Poskim hold that the Isur is only when one is Motzi others.

Kaf ha'Chayim (6): The Chayei Adam does not permit Tehilim. It is better to repeat a Mishnah 100 times than to say Tehilim by heart.

Kaf ha'Chayim (2,3,4): Poskim argue about whether the Isur is mid'Oraisa or mid'Rabanan. The Birkei Yosef says that it is mid'Rabanan; one may rely on this to be lenient in pressed circumstances, or to say word for word after someone reading from a Sefer. The Isur does not apply to Targum.

Kaf ha'Chayim (7): Whenever one says Hash-m's name it is best to have in mind the Kri (Aleph-Dalet, Master of all) and Kesiv (Yud-Kei, He always was, is, and will be). Therefore it is best to always learn from a Sefer, even things which everyone knows.