Gemara

(Mishnah): If a barrel of Terumah wine broke in the upper winepress and it is about to fall into Tamei Chulin below, if one cannot save a Revi'is in Taharah:

R. Eliezer says, he must allow the Terumah to fall [even though it will mix with the Chulin, become Tamei and forbid it all even to a Kohen]. He may not actively be Metamei it (e.g. catch it in a Tamei vessel);

R. Yehoshua says, [in order to save the Chulin] he may actively be Metamei the Terumah.

Eruvin 67b: A case occurred in which the hot water prepared for a baby spilled. Rabah told people 'bring hot water from my house'!

Abaye: There is no Eruv!

Rabah: Tell a Nochri to bring the water from my house.

Question (Abaye): Haza'ah (sprinkling Mei Chatas to be Metaher) is Shevus [forbidden mid'Rabanan on Shabbos], just like Amirah l'Nochri (telling him to do what a Yisrael may not do);

Just like we may not do Haza'ah on Shabbos [even if needed to bring Korban Pesach], similarly we may not tell a Nochri [to bring water without an Eruv for the sake of Milah]!

Answer (Rav Yosef): We distinguish Shevus with an action from Shevus without an action;

Rabah did not tell a Nochri to heat water!

Rishonim

Rif (Shabbos 56a) and Rosh (Shabbos 19:2): Amirah l'Nochri is Shevus when he tells him to do Melachah.

Ran (DH Aval): We do not permit Shevus d'Shevus to avoid a loss. This is why one may not tell a Nochri to extinguish, even though extinguishing is Shevus, for it is a Melachah she'Einah Tzerichah l'Gufah.

Rambam (Hilchos Shabbos 6:9): Something that is not a Melachah, and it is forbidden on Shabbos only due to Shevus, one may tell a Nochri to do it on Shabbos if there is slight illness, a big need, or for a Mitzvah.

Rebuttal (Ra'avad): If there is slight illness, the Yisrael himself may do by himself. One who is moaning may suckle [from an animal - Kesuvos 60a]. We may return a broken bone (Shabbos 148a). Similarly, to avoid a loss, e.g. [a backup of water due to] a clogged drainpipe, one may crush [what is clogging it] with his foot.

Rambam (10): A Yisrael may tell a Nochri to bring hot water from one Chatzer to another without an Eruv to bathe a child in pain, or all similar matters.

Mordechai (Shabbos 458): Shevus without an action is more lenient. Why did Chachamim forbid telling a Nochri to extinguish, even to avoid a loss? Several Isurim mid'Rabanan were permitted to avoid a loss, e.g. clearing a clogged drainpipe, causing Tum'ah to Terumah in order to save Chulin (Pesachim 15a), carrying a wallet less than four Amos at a time on Shabbos, or telling a Nochri to take it home, or letting blood from a Bechor (Pesachim 11a)! I say that in those cases, if we do not permit, he might do a big Isur, i.e. unclog it by hand, eat a mixture of Terumah [even though he is a Yisrael, or he or it is Tamei], or carry four Amos in Reshus ha'Rabim. If we would permit telling a Nochri to extinguish, the Yisrael extinguish himself, for people panic when fire approaches their property.

Terumas ha'Deshen (66): Or Zaru'a permits reheating Chullent through Nochrim, unless Rabanan are not found [lest people be even more lenient]. It must still be warm, k'Ma'achal Ben Drusai. Also Semag, the Mordechai and Bahag say so. They discussed for Chupos, but the leniency is even without Simchas Chasan and Kalah. A [Nochri] slave may put food on the stove used to heat the house, and light the stove, even though this also heats the food. If they intend for Heter, i.e. to heat the house, it is permitted even if it totally cooled off. One may tell a Nochri [to heat the house], since all are considered sick regarding the cold. Putting the food on the stove does nothing. When lighting the stove, the Nochri intends mainly to heat the house. Even though it is a Pesik Reishei regarding the food, for Shevus mid'Rabanan, we distinguish between real intent, and a Pesik Reishei without intent. The Mordechai brings so from Maharam. However, the Yisrael may not put the food on the stove, even before the Nochri lights it. One may not put food in front of a Nochri if he knows that the Nochri will take it outside the Chatzer.

Terumas ha'Deshen (63): A Gadol forbids making the forms of letters in the air, for he trains his hand to write. The Gemara said "if one saw a craftsman on Shabbos and learned, is he liable?!" For Shevus, we can distinguish doing an action from inaction.

Kaf ha'Chayim (307:42): The distinction between action and in action applies [also] to the Shevus of Amirah l'Nochri.

Poskim

Shulchan Aruch (OC 307:2): After Shabbos, one may ask a Nochri "why didn't you do this [Melachah] for me?", even though he understands that you want him to do it the next Shabbos.

Mishnah Berurah (11): The Yisrael may not benefit from the Melachah until after Shabbos, like the law of a Nochri who did Melachah for a Yisrael. Alternatively, the Yisrael's body does not benefit from the Melachah, e.g. there already was a Ner, and the Nochri added another Ner. If not, even if the Nochri decided himself to do the Melachah, a Yisrael may not benefit from it.

Shulchan Aruch (5): Something that is not a Melachah, and it is forbidden on Shabbos only due to Shevus, one may tell a Nochri to do it on Shabbos if there is slight illness, a big need, or for a Mitzvah.

Magen Avraham (7): For real illness, i.e. the entire body is affected, or there is danger to a limb, one opinion (328:17) permits even a Yisrael to do Shevus. One may do Shevus abnormally even for one in pain (238:33). It seems that the Magid Mishneh's and Tur's text of the Rambam said "slight illness or [slight] pain. It did not say "or a great need." It seems that this was in the Ra'avad's text. The Mordechai (Shabbos 458) permits a clogged pipe, so he will not come to fix it with his hands. Other Shevusim are forbidden. (334:26). The Rivash (384) forbids Amirah l'Nochri of Shevus, even to avoid a loss. Extinguishing is Melachah she'Einah Tzerichah l'Gufah [and one may not tell a Nochri to do it to avoid a loss]. If a Nochri brought a gift, one may not take it, even though he did only an Isur mid'Rabanan. In one answer, the Ran forbids Shevus in the normal way, even through a Nochri, to avoid a loss. Another answer says that extinguishing is different, for if it is Tzerichah l'Gufah, mid'Oraisa one is liable for it, and not everyone is expert about this. Therefore, we forbid even through a Nochri. In 334:2, he says that some forbid even Shevus d'Shevus, like the Rashba (784). I say that to avoid a big loss, one may permit Shevus through a Nochri even normally, and through a Yisrael abnormally. If not, one may not be lenient at all.

Mishnah Berurah (22): Eliyahu Rabah is unsure even about a big loss. See Sa'if 19 regarding rain that is ruining merchandise. (One may call a Nochri, even though he is sure that the Nochri will save the property.)

Kaf ha'Chayim (41): The Heter is only to avoid a big loss that came suddenly, and surely not if he knew before Shabbos.

Kaf ha'Chayim (38): Shev Yakov (3) defends the custom that a Nochri comes to the Beis ha'Keneses and extinguishes the Neros on Shabbos or Yom Tov night. He has other reasons.

Shevus Yakov (3:16): Since one may hire a Nochri b'Kablanus (he is paid one amount for the entire job) for needs of the Rabim [including Melachah them on Shabbos], all the more so it is permitted to avoid a loss to an individual.

Shulchan Aruch (ibid.): A Yisrael may tell a Nochri to bring water from one Chatzer to another without an Eruv to bathe someone in pain.

Gra (DH Devar): The Rambam holds that the episode was not on the day of the Milah. Rather, they wanted to bathe the baby merely due to pain. The same applies to a Mitzvah. Abaye asked from Haza'ah, and Rav Yosef distinguished between different kinds of Shevus. This is like the Rif and Rosh.

Beis Yosef (DH Kasav): Some permit Shevus d'Shevus only for proper illness [but not for a slight illness]. Avi ha'Ezri, brought in Mordechai Shabbos 474, forbids Amirah l'Nochri only for a Torah Isur that can never be done through a Yisrael, e.g. heating water [on Shabbos]. A Yisrael could carry in Reshus ha'Rabim, through a wall of people, so Amirah l'Nochri is permitted for this. If there is concern for confusion with a Reshus ha'Rabim, Amirah l'Nochri is forbidden even for a mid'Rabanan matter, even for a Mitzvah, like the water for Milah that spilled. Perhaps he permits [what could be permitted, e.g. through a wall of people] only for the sake of a Mitzvah. However, why does he forbid when it could be confused with a Reshus ha'Rabim? This contradicts what he just said! This requires investigation.

Answer (Taz 4): The Mordechai permits only when there are people here to make a wall. He forbids Karmelis when there are no people. Above, the Mordechai said that he did not command to bring water through Karmelis, rather, through roofs, Chatzeros and Karfifos.

Bi'ur Halachah (DH Lehavi): The same applies to bringing through Karmelis (YD 266:6). It is forbidden only through Reshus ha'Rabim. Eliyahu Rabah brings so from several Poskim, unlike Hagahos Mordechai, brought in the Taz, who forbids.

Shulchan Aruch (ibid.): Some forbid.

Gra (DH v'Yesh): Rav Ashi questioned why Ameimar had his eye painted through a Nochri on Shabbos (Beitzah 22a). This shows that even Shevus through a Nochri is permitted only for a Choleh, for whom we permit proper Shevus through a Yisrael]. See the Ramban [130b DH Ha].

Mishnah Berurah (23): The Levush and Eliyahu Rabah say that the Halachah follows the first opinion.

Kaf ha'Chayim (48): The Shulchan Aruch favors the first opinion, for he brought it Stam, and called this opinion 'some say.' The Rema agrees.

Rema: Below (586:21), he permits. See 276:2, that some permit even a Torah Isur.

Taz (4): Ba'al ha'Itur permits. The Rema wrote above that one should be stringent. This is primary, for Ba'al ha'Itur is an individual against Rabim.

Mishnah Berurah (24): Amirah l'Nochri is Shevus, even if he tells a Nochri to tell another Nochri to do Melachah on Shabbos (Avodas ha'Gershoni). Chavos Ya'ir (53) is lenient, and all the more so for a matter that is only Shevus. Sefer ha'Chayim says that one may rely on the lenient opinion to avoid a big loss.