Gemara

(Mishnah): The Kohen Gadol reads (Avodas Yom Kipur) in Acharei Mos and "Ach be'Asor".

Contradiction (Mishnah): One may skip from Parashah to Parashah when reading Nevi'im (b'Tzibur), but not when reading the Torah.

Answer: One may skip (when reading the Torah) only if the Torah can be rolled to the next Parashah within the time that the translator translates the last verse.

Objection (Beraisa): One may skip in Nevi'im, but not in Torah. The skip must be so short that the Sefer can be rolled to the next Parashah within the time that the translator translates the last verse.

This implies that even such skipping is not allowed in the Torah!

Answer (Abaye): One may skip to a related Parashah, not to an unrelated Parashah.

Support (Beraisa): One may skip in the Torah to a related Parashah, and in the Nevi'im, even to an unrelated Parashah. In both cases we must be able to roll the Sefer to the next Parashah within the time to translate the last verse;

We may not skip from one Navi to another. This is only allowed within the 12 minor Nevi'im, and only from an earlier Navi to a later one.

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.

Rishonim

The Rif (Megilah 14a) and Rosh (Megilah 3:9) bring our Gemara. They permit to skip to a related matter even if it is far away, such as from Acharei Mos to "Ach be'Asor".

Question: The Gemara says that this is permitted because they are close!

Answer (Bach OC 144 DH v'Teimah): The Rif and Rosh teach that one may skip even more than from Acharei Mos to "Ach be'Asor", as long as it is within the time to translate.

Rambam (Hilchos Tefilah 12:8): One may skip in the Torah to a related matter, such as from Acharei Mos to "Ach be'Asor". He may not read even one word by heart. One may skip on condition that he can roll the Sefer to the next Parashah within the time to translate the last verse.

Rambam (ibid. 13): In Navi one may skip to an unrelated Parashah.

One may not skip from one Navi to another. It is permitted only in Trei Asar. One may not skip from the end of the Sefer to the beginning. One may not skip longer than it takes to roll the Sefer to the next Parashah.

Kesef Mishneh: Tosfos derives that one may not go from a later Navi to an earlier Navi. It cannot mean from the very end to the beginning, for this is more than the time to translate, and it is forbidden even from beginning to end. This refers to skipping from one Navi of Trei Asar to another. Within one Navi one may skip backwards, just like in other Nevi'im.

Rebuttal (Magen Avraham 144:4): Also in other Nevi'im one may not skip backwards, like it says in Minhagim. Perhaps the Kesef Mishneh holds that it is permitted within one Parashah.

Poskim

Shulchan Aruch (OC 144:1): One may skip in Navi from one Parashah to an unrelated Parashah.

Mishnah Berurah (1): Letter of the law, the Haftorah was not fixed. The only requirement is to read something relevant to the Torah Parashah. It is permitted to skip to different Parashiyos. Our custom is to add verses in other places in several Haftoros.

Shulchan Aruch: One may not skip in Torah to an unrelated Parashah. This is because those listening will be confused. One may skip (even in Torah) to a related matter, such as from Acharei Mos to "Ach be'Asor", which the Kohen Gadol reads on Yom Kipur. But he may not read even one word by heart.

Gra (DH v'Hu): The Kohen Gadol reads by heart because there is no alternative.

Kaf ha'Chayim (3): The Eshel Avraham (1) says that on a Ta'anis Tzibur we skip to "Pesal Lecha", like the Kohen Gadol skipped from Acharei Mos to "Ach be'Asor". Alternatively, we are not concerned when a second person reads the second Parashah. The custom is to be stringent like the second reason. Therefore, if the Kohen will also read the second Parashah he should not skip.

Shulchan Aruch (ibid.): In Navi one may skip to an unrelated Parashah, as long as the Tzibur will not need to wait in silence. This is within one Navi. We may not skip from one Navi to another.

Question (Beis Yosef): Today the custom is to append to the Haftorah two or three verses from "Sos Asis" in Yeshayahu when there is a Chasan, and verses from "Machar Chodesh" when Shabbos is Erev Rosh Chodesh. This is from Navi to Navi!

Answer #1 (Terumas ha'Deshen cited in Beis Yosef ibid.): The Isur is because it is dishonorable for the Tzibur to wait. This was when Nevi'im were written on rolled parchments. Nowadays we read the Haftorah from a book. It is easy to put a marker so one will find the place very quickly, so one may skip from Navi to Navi. However, Rashi says that we do not skip because those listening will be confused. We do not learn Halachos from Navi, therefore we are not concerned for if people are slightly disoriented. According to Rashi we do not skip from Navi to Navi, for this greatly confuses people. This applies even nowadays. The Shi'ur for skipping applies only where they translate. Our custom is not to translate.

Answer #2 (R. Mano'ach cited in Beis Yosef ibid.): Since we say "Sos Asis" by heart, we are not concerned for skipping. The Nimukei Yosef says that it is not considered part of the Haftorah, rather, like a mere song.

Shulchan Aruch: One may skip from one Navi to another in Trei Asar as long as he does not skip from the end of the Sefer to the beginning.

Mishnah Berurah (8): Trei Asar is considered like one Navi because they are short. Therefore, it does not befuddle people if we skip from Navi to Navi.

Shulchan Aruch (3): We do not roll a Sefer Torah b'Tzibur for it is dishonorable to the Tzibur. If they have only one Sefer Torah and they need to read in two places, they roll; the honor of the Tzibur is overridden.

Beis Yosef (DH v'Chosav ha'Mordechai): The Rashba says that it a Tzibur has only one Sefer Torah and they need to read in two places, they may read the second Parashah in a (printed) Chumash. The Kohen Gadol was allowed to read by heart because of the honor of the Tzibur, all the more so one may read from a Chumash for this reason! Our custom is like the Mordechai and Ritva, who say that we override the honor of the Tzibur, but we do not neglect the enactment of Chachamim (to read both Parashiyos from a Sefer Torah).

Kaf ha'Chayim (17): The Rashba holds that we read from a Chumash without a Berachah.

Magen Avraham (7): In the Mikdash we do not override the honor of the Tzibur because all of Yisrael are there. There are a small number of people in the Beis ha'Keneses, they pardon their honor so they can fulfill the enactment of the Ge'onim.

Darchei Moshe (1): We may roll while people are saying Pesukei d'Zimra. If we must roll during Kri'as ha'Torah, we do so.

Mishnah Berurah (13): The Magen Avraham (6) permits to roll one Sefer while someone is reading in the other. It is better to do like the Darchei Moshe.

Kaf ha'Chayim (13): If someone removed a Sefer Torah from the Aron and then realized that a different Sefer is rolled to where they must read, Poskim argue about whether it is better to return it and take the other, or to roll and use the first to avoid disgrace to it. Most favor the latter opinion.

Kaf ha'Chayim (15): When we read from three Sifrei Torah, if there are only two Seforim in the Beis ha'Keneses we read the third Parashah in the first Sefer used.