Gemara

(Beraisa - Tudus) Question: Why did Chananyah, Misha'el and Azaryah [want to] enter the furnace and give up their lives?

Answer: They made a Kal va'Chomer from the plague of frogs. Even though they are not commanded about Kidush Hash-m, "u'Va'u b'Veisecha uv'Sanurecha uv'Mish'arosecha" (they entered ovens and kneading bowls);

Kneading bowls are near ovens when the ovens are hot, and still they entered. We are commanded about Kidush Hash-m, all the more so we should enter!

Avodah Zarah 27a: If one's brothers died through circumcision [he is exempt from circumcision due to the danger].

Shabbos 21b (Beraisa): The Mitzvah is to leave Ner Chanukah outside the door of one's house. At a time of danger, it suffices to leave it on one's table.

130a (Beraisa - R. Shimon ben Elazar): Every Mitzvah that Yisraelim were willing to die for when kingdoms decreed against it, like idolatry and Milah, they still solidly observe it. Every Mitzvah that they were not willing to die for, like Tefilin, their observance of it is weak.

Romi decreed to kill anyone who will wear Tefilin. "Elisha of the wings" wore Tefilin in the market. An officer chased him and caught him. Elisha took the Tefilin from his head and put it in his hand.

Elisha said that it is only dove's wings. A miracle occurred. He opened his hand and revealed dove's wings.

Rishonim

Rambam (Hilchos Yesodei ha'Torah 5:4): Whenever the Halachah is to transgress and not be killed, if one was killed and did not transgress, he is liable for his own death.

Rambam (Hilchos Rotze'ach 1:14): Anyone who could save (someone being pursued), but did not, transgressed "Lo Sa'amod Al Dam Re'echa."

Kesef Mishneh: Hagahos Maimoniyos says that the Yerushalmi (Terumos 8:4) concludes that one must enter Safek danger to save another. It seems that this is because the other is in definite danger.

Bach (CM 426:2): The Beraisa connotes that even if it is not clear that he will be able to save, he must enter Safek danger to try. However, the Rambam wrote 'he could save', which connotes that he surely can save. This is why the Tur says 'and the Rambam says...', for he rules unlike this.

Rambam (7:8): If a Shogeg murderer went to exile, even if all Yisrael need him, like Yo'av, he does not leave until the Kohen Gadol dies.

Or Some'ach: The murderer should not put himself in Safek danger to save another from Vadai danger, unlike the Kesef Mishneh cited from the Yerushalmi. Really, there is no proof from the Yerushalmi.

Poskim

Shulchan Aruch (YD 157:1): If a Nochri threatens a Yisrael to transgress a Mitzvah or be killed, for any Mitzvah other than idolatry, Arayos and murder, if it is in private, the Yisrael transgresses and is not killed. If the Nochri intends to make the Yisrael transgress, the Yisrael may be stringent and submit to be killed.

Beis Yosef (DH u'Mah): The Tur says that for other Arayos (for which one need not be Moser Nefesh), one may be stringent to be killed and not transgress. This is like Semak, the Ran (Shabbos 22b), and Rosh (Avodah Zarah 2:9). R. Yerucham says that this is only when the Nochri intends to make the Yisrael transgress, but not if he intends for his own benefit. The Rambam says that he is liable for his own death; it seems that the Ramban agrees. The Nimukei Yosef (Sanhedrin 18a) wrote like the Rambam, but says that a great Chasid with Yir'as Shamayim may be stringent to be Moser Nefesh even for a light Mitzvah if he sees that the generation rampantly transgresses the matter. The Gemara answered that Chananyah, Misha'el and Azaryah were Moser Nefesh because people thought that the image was idolatry, so there was Kidush Hash-m. Vayikra Rabah (32:1) records people who were killed for Bris Milah or Netilas Lulav. This was Midas Chasidus.

Shach (1): The Bach rules like the Rambam, who says that anyone who should have transgressed, but chose to forfeit his life, is liable for his own death. The Terumas ha'Deshen says that normally, we are lenient about Safek Nefashos. However, regarding Kidush Hash-m the Torah was not concerned for loss of life, so we cannot learn from other Sefekos. We judge according to the case and his intent.

Note: The Shach himself (7) is lenient about a Safek whether or not one must be Moser Nefesh at a time of decrees if the Nochri intends for his own benefit.

Gra (3): We learn from Shabbos 130a. (Mitzvos that Yisraelim were willing to die for, they still solidly observe them.) Elisha, R. Akiva, R. Yehudah ben Beseirah and all those killed by the kingdom were Moser Nefesh for individual Mitzvos, such as observing Shabbos, eating Matzah, and Bris Milah. The Rambam must say that all of these were necessary at the time. Chananyah, Misha'el and Azaryah were Moser Nefesh because it was in public. Still, it is difficult, for it was for the Nochrim's benefit. Daniel was Moser Nefesh for Tefilah in his house (Daniel 6:8-11).

Shach (YD 157:3): If there is risk to a limb if one will not transgress a Lav, perhaps this is like loss of money (so one may not transgress), or like loss of life (one may transgress). It seems that we are lenient.

Rema: One must protest [against transgressors]. If not, he is held responsible. If this could be dangerous, he need not spend money for this.

Source (Gra 5): Chachamim did not reveal families with lineage problems, for it would have been dangerous (Kidushin 71a). If not for a verse, we would have said that one need not even pay money to save another (Sanhedrin 73a)!

Pischei Teshuvah (5): Below (334:48), the Rema says that one need not spend money to protest, even if there is no danger. The Levush here says so.

Shulchan Aruch (ibid.): If the Nochri intends for his own benefit, one transgresses and he is not killed. If there is a decree [only against Yisrael], one must be Moser Nefesh even for [the Yisrael custom of] shoelaces.

Rema: This is only if they want the Yisrael to transgress a Lav. If they decreed against an Aseh, one need not fulfill it and be killed. However, if it is needed at the time, one may fulfill it and be killed.

Gra (10): Elisha took the Tefilin off his head, because the Nochrim could do so themselves and physically stop him from doing the Mitzvah. This is the meaning of Karka Olam. The Ran says that Elisha and Daniel chose to be stringent. The Nimukei Yosef says that we must say that it was needed at the time. Above, the Shulchan Aruch ruled unlike the Nimukei Yosef. Why was the Rema quiet there?

R. Akiva Eiger (DH v'Lo): The Radvaz (2:145) says that if one is among Nochrim, and all the food is forbidden, even mid'Rabanan, he must spend all his money to get Kosher food, and not eat Isur. After his money runs out, this is Ones. He eats the least severe Isur. He need not borrow money [in order to buy more Kosher food], lest he be unable to repay, and the Nochrim imprison and afflict for this, and he can come to danger.

Shulchan Aruch (CM 426:1): If Reuven sees Shimon drowning..., and Reuven himself could have saved him or hired others to save him, but he did not, he transgressed "Lo Sa'amod Al Dam Re'echa."

SMA (2): Hagahos Maimoniyos obligates entering Safek danger to save another in definite danger. Perhaps the Shulchan Aruch and Rema omitted this because the Rif, Rambam and Rosh did not bring it.

Radvaz (3:627 (1052), cited in Mishbetzos Zahav OC 328:7): One must use his money to save others, but he need not endanger his limbs. If it is mortal danger, he is a crazy Chasid. The Safek of his life overrides the certain death of his friend.

Yam Shel Shlomo (Bava Kama 10:50): Rav Kahana killed a Moser (Bava Kama 117b). Rav told him to exile himself. He was allowed to kill the Moser; he repented for acting in front of his Rebbi without asking. Also, he endangered himself. The Yevanim did not tolerate murder; he should not have relied on a miracle.