Gemara

121a (Mishnah): If a Nochri comes to extinguish, [the owners of the burning property] do not (i.e. may not) tell him to extinguish, nor do (must) they tell him not to extinguish, because his rest on Shabbos is not their responsibility.

(R. Ami): Regarding a fire, Chachamim permitted to say [in front of Nochrim] 'anyone who extinguishes will not lose!'

(Beraisa): Once, a fire erupted in the Chatzer of Yosef ben Simai. The [Nochri] authorities of Tzipori came to extinguish it, for Yosef was the overseer of the king's money. He did not allow them, due to the honor of Shabbos [lest people think that he asked them to extinguish];

A miracle occurred, and rain fell and extinguished it.

At night, he sent money to everyone [who came to extinguish but consented to refrain], and to the mayor.

Chachamim: He did not need to stop them. It was taught that if a Nochri comes to extinguish, they do not tell him to extinguish, or not to extinguish.

(Mishnah): If a minor comes to extinguish, they do not allow him, for it is their responsibility that he rest.

(R. Yochanan): The case is, the child comes to extinguish because the father wants this.

Question: If so, likewise the Nochri [in the Reisha] does so for the sake of the Yisrael. However, this is forbidden!

Answer: A Nochri acts for his own sake.

150a (Mishnah): [On Shabbos,] one may not ask his friend to hire workers for him.

(Rav Papa): The Mishnah forbids asking a Nochri friend.

Objection (Rav Ashi): It is Shevus (forbidden mid'Rabanan) to ask a Nochri to do things forbidden on Shabbos!

153a (Mishnah): If one was on the road when Shabbos was about to come, he may give his wallet to a Nochri [before Shabbos, to take it home for him];

Chachamim know that a person cannot control himself if he stands to lose much money. If we do not allow him a way to save it, he might carry it four Amos in Reshus ha'Rabim!

Kesuvos 70b (R. Ami): Regarding a fire [on Shabbos], one may say 'whoever extinguishes will not lose.'

This is permitted regarding a fire, but not for other Isurim of Shabbos.

Beitzah 35b (Mishnah): One may lower Peros into the house through a skylight on Yom Tov, but not on Shabbos. One may cover Peros with Kelim to save them from a drip.

36a (Ula): One may cover even bricks.

Question: The Mishnah permits only Peros!

Answer: Since the Reisha discusses lowering Peros, the Seifa discusses covering Peros.

(R. Yitzchak): One may cover only Peros proper [to benefit from them today].

This is like R. Yitzchak taught elsewhere, that one may move a Kli only for the sake of something that one may move on Shabbos.

Rishonim

Rambam (Hilchos Shabbos 12:7): Regarding a fire, Chachamim permitted to say 'whoever extinguishes will not lose.'

Rosh (16:10): Bahag says that just like one may say [in front of Nochrim] 'anyone who extinguishes will not lose', the same applies to water that broke through a dam [and is about to damage]. It seems that the same applies to any sudden damage, e.g. a barrel of wine faltered and is about to break. One may call a Nochri, and when he comes, automatically he will extinguish. One may even tell him 'whoever extinguishes will not lose.' All the more so one may call him!

Shiltei ha'Giborim (45a:4): It seems that the Rambam in Perek 11 (surely this should say Perek 12 - PF) disagrees with Bahag. Also Semag and Hagahos Maimoniyos permit only for a fire. Mahari'ach agrees with the Rosh. He says that perhaps a fire is different than other Isurei Shabbos. For a fire, one may tell an individual 'if you extinguish, you will not lose.' For other Isurim, one may say only Stam ['whoever saves will not lose.']

Question: If rain came and it will ruin merchandise, may one move it or tell a Nochri to move it? Sefer ha'Terumah permits to hide coins, due to the authorities.

Answer (Rashba 1:784): Sefer ha'Terumah is wrong. Only regarding a fire, Chachamim permitted to tell a Nochri 'whoever extinguishes will not lose.' The Gemara says 'for a fire they permitted.' This excludes other Isurim of Shabbos. We learn also from Beitzah 36b. A drip was ruining Abaye's mill. Rabah told him to move his bed there, to make it [like] Geraf Shel Re'i (a Kli for excrement. One may move it if it bothers him. One may not move a mill just to avoid a loss.) If one was on the road when Shabbos came, he may give his wallet to a Nochri (153a). There Chachamim permitted, for if we do not permit, he may come to do a Torah Melachah. Here, why should we permit moving? Even telling a Nochri is permitted only to evade an Isur Torah. Normally, whatever one may not do, he may not tell a Nochri to do it.

Poskim

Shulchan Aruch (OC 307:19): If rain or something else is ruining merchandise, or a river or swamp of water is coming to damage property, or a barrel of wine faltered and is about to break, one may call a Nochri, even though he is sure that the Nochri will save the property.

Mishnah Berurah (68): One may call a Nochri even if he will need to do a Torah Melachah, e.g. carry from Reshus ha'Yachid to Reshus ha'Rabim. One may overtly command a Nochri to carry Muktzeh, or from Reshus ha'Yachid to Karmelis, if there will be a big loss, based on Sa'if 5.

Mishnah Berurah (69): Since a person is frantic about his money, if we do not permit in these ways, he may come to do a Torah Melachah to save it.

Shulchan Aruch (ibid.): Similarly, one may tell the Nochri 'whoever saves will not lose'', like they permitted regarding a fire to say "whoever extinguishes will not lose."

Beis Yosef (DH Kasav): The Agur holds like the Rashba, that one may say "whoever extinguishes will not lose" only for a fire. Semag and Sefer ha'Terumah prove this from Kesuvos 70b, unlike Bahag, who equates other Isurim of Shabbos to a fire. Semak says similarly, but concludes "perhaps Bahag explains that the Gemara discusses when there is no loss." The Rosh rules like Bahag. The Tur brought this in 334. Since this is a mid'Rabanan law, one may rely on the Rosh and Bahag.

Mishnah Berurah (70): One may not say in the second person "if you save, you will not lose." Eliyahu Rabah (43) brings so from Shiltei ha'Giborim. YD 221:8 connotes like this. (If a vow forbids Reuven to benefit from Shimon, and Reuven needs food, Shimon may say "whoever feeds Reuven will not lose", but he may not say "if you feed Reuven, you will not lose.")

Kaf ha'Chayim (129): For a fire one may say "if you extinguish, you will not lose" (Eliyahu Rabah).

Kaf ha'Chayim (128): He will not lose the wages for his toil.

Shulchan Aruch (ibid.): Some say that they permitted only for a fire.

Magen Avraham (27): One who is lenient does not lose.

Mishnah Berurah (71): Below (334:26), the Shulchan Aruch brings only the lenient opinion. All permit to call a Nochri and show him, without telling him anything.

Kaf ha'Chayim (131): If a Nochri (Tony) owes David, and the Nochri was sentenced to be executed today on Shabbos, Beis Yehudah (12) permits David to ask another Nochri (Peter) to encourage Tony to admit to any debts he has, but Peter should not explicitly mention the debt to David.

Erech Lechem: If one has no food to feed a child, one may tell a Nochri to make a food made from flour and water, for regarding hunger, a child is like a sick person without danger.