Gemara

38b (Rav Yehudah): One who frees a slave transgresses an Aseh - "forever you will make them work for you."

Question (Beraisa): A case occurred in which R. Eliezer found only nine men in a Beis ha'Keneses, and he freed his slave to complete the Minyan.

Answer: Rav Yehudah agrees that one may free a slave if needed for a Mitzvah.

38a: Men were sinning with a Shifchah in Pumbadisa.

Abaye: If not for Rav Yehudah's law, I would force her master to free her.

Ravina: Rav Yehudah agrees that one may free a slave to prevent transgression!

Question: Surely, Abaye agrees! R. Chinena bar Rav Ketina taught that a case occurred in which Chachamim forced a master to free his half-Shifchah. Rav Nachman bar Yitzchak explained that men were freely sinning with her.

Answer: One may free a half-Shifchah, for there is no other way to stop transgression. She may not marry a slave, nor a free men. One can pair a Shifchah with a male slave; he will stop others from sinning with her!

41a (Mishnah - Beis Shamai): A half-slave cannot marry a Shifchah, nor a Bas Yisrael. "Lasheves Yetzarah" (the world was created for procreation)! Rather, for Tikun ha'Olam, we force his master to free him, and the half-slave writes a document obligating himself to pay half his value to his master.

Beis Hillel agreed to Beis Shamai.

Shabbos 4a (Rav Bivi bar Abaye) Question: If Reuven stuck a dough in an oven on Shabbos, may he remove it before it bakes, to avoid a Chiyuv Chatas?

(Rav Shila): Rav Bivi really asked whether or not someone else (who knows that Reuven put it in b'Shogeg) may remove it to exempt him.

Objection: One may not sin (even mid'Rabanan) to save someone else from sin!

Eruvin 32a (Beraisa): If Reuven (a Chaver, i.e. someone reliable about Ma'aser) told Shimon 'go fill this basket with dates from my tree', he may eat without tithing (surely, Reuven took Terumah);

R. Shimon ben Gamliel says, he must tithe them. A Chaver is not suspected to Torem Lo Min ha'Mukaf (when the Terumah is not near the produce.)

Rebbi holds that it is better for a Chaver to transgress a minor matter in order that an Am ha'Aretz not transgress a severe matter. R. Shimon disagrees.

103b (Beraisa): (On Shabbos) if there is a wart on a Kohen, another Kohen tears it off with his teeth (to enable the former to serve).

Pesachim 59a: The only Korbanos offered after the afternoon Tamid are Pesach and that of a Mechusar Kipurim (one who must offer birds before eating Kodshim, e.g. a Zav) on Erev Pesach. The Mitzvas Aseh of Korban Pesach has Kares (for one who does not bring it). To enable one to fulfill it, we override the Aseh of Hashlamah (not to offer after the afternoon Tamid), which does not have Kares;

Rishonim

Tosfos (Shabbos 4a DH v'Chi): Our Gemara forbids even a light sin to save another from a big sin. Rebbi holds that it is better for a Chaver to violate a minor matter to stop an Am ha'Aretz from violating a severe matter due to the Chaver. If the sin is not due to the Chaver, Rebbi agrees that the Chaver should not sin at all. For the great Mitzvah of Peru u'Rvu, we force a master to free his half-slave. R. Eliezer freed a slave for the sake of a Mitzvah of the Rabim. Further, we can say that one may not do a light sin to save someone else who transgressed from a bigger sin (but one may to save someone blameless). This explains why Kohanim may offer for a Mechusar Kipurim after the Tamid on Erev Pesach, and a Kohen may tear off a wart on another Kohen to enable him to serve. They forced a master to free his half-Shifchah because she made herself available for Zenus, and it is as if men were coerced, so it was like a Mitzvah of the Rabim.

Poskim

Shulchan Aruch (OC 306:14): If one heard that Nochrim took his daughter to uproot her from Yisrael, it is a Mitzvah to pursue to strive to save her. He may even go past three Parsa'os. If he does not want, Beis Din obligates him.

Beis Yosef (DH Nish'al...): The Rashba (7:267) was unsure whether the opinion that the Techum of three Parsa'os is mid'Oraisa permits this, and whether one may bring documents to save her. He leaned to forbid, for Levi may not violate even a light matter to save Ploni from a big sin, unless it is due to Levi. Tosfos permits for the sake of a big Mitzvah, or when Ploni is blameless. Both answers permit here. If he does not want, we force him, like we force a master to free a half-slave. It seems that we permit even Chilul Shabbos mid'Oraisa. This is a small Isur compared to a girl becoming an apostate and transgressing her entire life.

Question (Bach DH Kosav): The Rema references Siman 328:10, in which he rules like the Rashba, who forbids one to be Mechalel Shabbos to save another who is being coerced to be Mechalel Shabbos. There, the Rema references here, in which the Shulchan Aruch rules like Tosfos. This connotes that the Halachah follows both of these!

Answer #1 (Bach): Here, we fear lest Nochrim scare her into becoming an apostate and sinning. There, he will be forced to transgress. Even if he serves idolatry, this is not an Aveirah. The Torah exempts Ones! We do not even change an enactment of Chachamim (that virgins get married on Wednesday) to avoid Ones (being raped by the governor)!

Rebuttal (Magen Avraham 29): We are more lenient to save one who was Ones than one who transgressed b'Mezid! The Rema (329:8) says that also Siman 306 discusses Ones!

Answer #2 (Magen Avraham): The Beis Yosef permits here because one Chilul Shabbos is small compared to a lifetime of sin. One may not desecrate Shabbos to prevent one sin, even idolatry, for Chilul Shabbos is like idolatry.

Note: The Taz (5) combines both answers: there he will sin only once b'Ones, and here she will be an apostate forever. I cannot infer his opinion when they seek to make someone transgress forever b'Ones, or only once b'Mezid.

Answer #3 (Gra DH Mi): We permit only when both answers of Tosfos apply. It is not a great Mitzvah to save someone who will be forced to sin only once.

Mishnah Berurah (56,57): If she transgressed, one may not save her. Eliyahu Rabah permits regarding an Isur mid'Rabanan. If she is blameless, one may transgress Melachos mid'Oraisa to save her.

Kaf ha'Chayim (106): The Levush permits saving other relatives. Other Acharonim permit saving even strangers. It seems that we force him to do so, but Mishbetzos Zahav says that we force only to save a daughter.

Magen Avraham (20): It is not clear whether or not we transgress for the sake of a minor. The Rashba does not allow an individual to save the Tzibur, but all the Poskim disagree.

Mishnah Berurah (57): Eliyahu Rabah permits saving a minor, for if not she will be like a Nochris her entire life.