Gemara

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 [in the Chatzer] or Shituf [in the Mavoy]!

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

Question (Abaye): Haza'ah (sprinkling Mei Chatas to Metaher) is Shevus [forbidden mid'Rabanan on Shabbos], just like Amirah l'Nochri (telling him to do something forbidden to Yisraelim];

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!

A case occurred in which the hot water prepared for a baby spilled. Rava told people to ask the mother if she would like hot water [to drink]. If she does, they may ask a Nochri to heat for her and the baby

91a (Beraisa - R. Yehudah): At a time of danger we used to take a Sefer Torah from Chatzer to roof, from roof to Chatzer and from Chatzer to Karfef to read from it;

Chachamim: You cannot bring a proof from a time of danger.

Rishonim

Rif (Shabbos 55b): If the hot water spilled after the Bris, we heat up more, due to danger. Bahag says that if they brought a knife before Shabbos, but it was stolen or dented, before the Bris, they may tell a Nochri to make or bring another knife.

Rif (ibid.) and Rosh (Shabbos 19:2): Rav Yosef distinguishes Shevus in which one tells the Nochri to do a Melachah, from an Amirah without a Melachah. Amirah l'Nochri is Shevus when he tells him to do Melachah. One may not tell a Nochri to make a knife or bring one through Reshus ha'Rabim.

Ran (DH v'Nimtza): Bahag permits Amirah l'Nochri even for a proper Melachah for Machshirei [matters needed to enable] Milah. If we would equate other Mitzvos to Milah, like the Rambam does, according to his opinion, we would permit Amirah l'Nochri of Melachah for all Mitzvos. It seems that Ba'al ha'Itur, who permits telling a Nochri to light a lamp for the sake of Seudas Shabbos, holds like this.

Ran (DH Aval): The Rif permits Amirah l'Nochri only for matters that if a Yisrael would do them, they are only Shevus. The Rambam (Hilchos Milah 2:9) agrees. He permits Shevus d'Shevus, i.e. Amirah l'Nochri to do Shevus, for Machshirei Lulav and Shofar. The Ramban permits only for Milah, since it overrides Shabbos. Other Mitzvos do not override Shabbos, so they do not override even Shevus d'Shevus, for 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 for a Mitzvah.

Rambam (10): A Yisrael may tell a Nochri to climb a tree or swim in order to bring a Shofar or a knife for Milah, or to bring from one Chatzer to another without an Eruv.

Tosfos (Gitin 8b DH Af): Even though Amirah l'Nochri is Shevus, it is permitted for the sake of Yishuv (settling) Eretz Yisrael. They do not permit Amirah l'Nochri of Melachah mid'Oraisa for other Mitzvos. One may not tell him to do an Isur Torah. One may not tell a Nochri to bring a Sefer Torah through Karmelis. We permit only for the sake of Milah, for Milah itself overrides Shabbos. Bahag permits telling a Nochri to do a Torah Isur for the sake of Milah. He must say that she was within seven days [and a Yisrael would heat for her]. However, in such a case one may not heat extra! We decree that healthy people may not eat what was cooked for a Choleh on Shabbos, lest they cook extra! Perhaps one may heat extra only for Bris Milah, but not for other needs, even through a Nochri.

Poskim

Shulchan Aruch (OC 307:5): 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 for a Mitzvah.

Beis Yosef (DH Kasav): The Magid Mishneh says that the Ge'onim agree that Machshirei Milah override Shevus d'Shevus, even though Machshirei Milah are not Docheh Shabbos. The Rambam learns to all Mitzvos. Some permit only for Milah, for it overrides Shabbos. Avi ha'Ezri, brought in Mordechai Shabbos 474, holds like the Rambam.

Mishnah Berurah (19): One may not tell a Nochri to do a Melachah even for a Mitzvah (331:6), e.g. to light a lamp to enable learning or praying.

Kaf ha'Chayim (39): In some places, when Yom Tov is on Motza'ei Shabbos, or on Yom Kipur before Ne'ilah, they tell a Nochri to light Neros in the Beis ha'Keneses. Birkei Yosef (514:3) said that if he could, he would stop this, for most Poskim do not permit proper Melachah. The Acharonim were stringent, and said not to rely on the lenient opinion even for a great need (276:2), especially for us [Sefardim], who accepted the Mechaber's rulings. The Mechaber permits Amirah l'Nochri only for Shevus. The Birkei Yosef solidly rejected those who defend the custom. Also the Noda bi'Yehudah (OC 34) denounced this, and abolished the custom in his locale. However, the Ben Ish Chai (Rav Pe'alim 2:43) gave ways in which it is permitted.

Kaf ha'Chayim (44): The Heter is only for a Mitzvah that day, but not for tomorrow (Noda bi'Yehudah 2 OC 44). Geza Yishai disagrees.

Kaf ha'Chayim (45): Once, the Esrog of a congregation became Pasul, and they told a Nochri to bring one from outside the Techum on Yom Tov. Devar Moshe (OC 36) distinguished between the Techum of one Mil and 12 Mil [which some say is mid'Oraisa]. He concluded that surely, one may tell a Nochri to bring it.

Shulchan Aruch (ibid.): A Yisrael may tell a Nochri to climb a tree to bring a Shofar for the Mitzvah. Some forbid.

Beis Yosef (DH Aval): The Tur holds like Tosfos. In Siman 338, he said that Avi ha'Ezri permits telling a Nochri to play instruments at a Chupah, but the Tur permits only for Mitzvos that override Shabbos. It seems that the Rashba (1:297) agrees. He says that we do not compare Shevusim. Chachamim did not allow Amirah l'Nochri of Melachah for the sake of Milah, but they permitted it for buying a house in Eretz Yisrael (310:4)! Only what Chachamim permitted is permitted.

Magen Avraham (8): This opinion permits Shevus d'Shevus only for Bris Milah, which overrides Shabbos. On 91a, we say that they used to take a Sefer Torah from the roof to the Chatzer to read in it. This shows that one may carry outside the Eruv for Keri'as ha'Torah! We cannot compare Shevusim to each other.

Gra (DH v'Yesh): Since we needed to say that it was permitted for Yishuv Eretz Yisrael, this shows that it is forbidden for other Mitzvos. The distinction between different Shevusim is only regarding Bris Milah.

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

Kaf ha'Chayim (46): Maharil (Teshuvah 116) forbids telling a Nochri to bring hot food or beer from one house to the house in which they are eating. It seems that if there is no Reshus ha'Rabim in between, the Mechaber permits, for this is Shevus d'Shevus for the sake of a Mitzvah.

Kaf ha'Chayim (49): Shirei Keneses ha'Gedolah says that the custom is to send through a Nochri for Simchos, Cholim or mourners. Honoring Chasan and Kalah is a Mitzvah.

Kaf ha'Chayim (52): Hosting guests from the field is a Mitzvah. Inviting guests from one's city is not. Some are lenient. Seudas Shabbos, or Seudas Chasan and Kalah, is a Mitzvah.

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

Mishnah Berurah (24): They are lenient for a Torah Melachah through a Nochri for the sake of a Mitzvah. The Rema wrote that one should be stringent, for this is a lone opinion. The Rif, Rosh, Rambam, and many other great Rishonim disagree. Only for Milah, the Magen Avraham (331:5) leans to rely on this opinion.