Gemara

(Rava): All babies are considered sick (need hot water) regarding Milah.

Shabbos 107a (Mishnah): If one pierces an abscess [just] to let out the pus, he is exempt.

Question: What is the source that it is permitted?

Answer (Mishnah): One may move a hand needle on Shabbos, for it is useful to remove a thorn.

129a (Rav Yehudah): We may make a bonfire for a Yoledes on Shabbos.

This is permitted also for a Choleh, in winter and summer.

(R. Chiya bar Avin): If someone let blood and [on Shabbos] got chilled, we may make a bonfire for him, even in summer.

Rishonim

Rambam (Hilchos Shabbos 2:14): We may make a bonfire for a Yoledes, even in summer, for the cold is very bad for her. One may not make a bonfire for a Choleh to warm himself. If one let blood and became chilled, we may make a bonfire for him even in Tamuz.

Rebuttal (Ra'avad): A Beraisa (Yoma 84b) explicitly permits warming water for a Choleh, whether for drinking or bathing. Perhaps this is when he is dangerously sick.

Hagahos Maimoniyos (6:5): If a fire was burning, and a Nochri added to it, a Yisrael may benefit from it, since initially he could have benefited a little. Similarly, if a Nochri made a fire for a child, or to make food for a child or for a Choleh without danger, a healthy person may benefit from it.

Hagahos Maimoniyos (6): If a winter room was warmed for minors or [Nochri] slaves who do not want to sit in the cold, adults may enter and benefit. R. Yom Tov permitted even for the sake of adults. We tell a Nochri to do needs of a Choleh without danger. All are sick regarding the cold. Maharam says that in France, they permitted in Rabbeinu's house. The Ri of Orlins permitted even [to tell a Nochri] to make the fire. However, in our kingdom it seems [proper] to forbid. He himself was stringent, and made a wall around the stove so that the slave cannot do anything. He said that this is permitted, but people are stringent to consider it forbidden to sit in a room heated on Shabbos. (One may not be lenient in front of them.) I remember being with my Rebbi in the Vasbork tower. We made a bonfire for warmth. Before it extinguished, slaves increased it, and said that they did so to benefit us. We sat and enjoyed it.

Hagahos Mordechai (Shabbos 472, very bottom of p.155): Slaves make bonfires on Shabbos in their masters' houses. Chachamim permitted only if the majority [of the wood] caught fire before Shabbos. We can say that a Nochri extinguishes for himself, but he may not burn a Yisrael's wood without permission. If the Yisrael saw and was silent, this shows that he consents. It is forbidden even in a Nochri's house, and all the more so in the Yisrael's house, for the Melachah is called on the Yisrael's name. The Yerushalmi says that one need not expel a Nochri from [doing Melachah in] the Nochri's house. This implies that one must expel a Nochri from the Yisrael's house.

Mordechai (ibid.): R. Yom Tov asked why people refrain from warming themselves near a fire that a Nochri made for Yisre'elim. R. Meshulam and all the Gedolim did so (did not refrain)! I say that just like we say that all are sick regarding Milah, all are sick, or at least pained, by the cold. One may suckle on Shabbos, which is Shevus, i.e. Mefarek k'Leacher Yad, due to pain. Even Aba Sha'ul (Yevamos 114a) forbids only Shevus with an action. We should permit even telling a Nochri, and all the more so if a Nochri did so by himself! One should not be stringent when there is pain. A Yisrael may have his eyes painted through a Nochri, even though he opens and closes his eye to help. We permit putting hot metal into the Mikveh due to the pain of the Kohen Gadol (Yoma 34b), even though this extinguishes a metal coal! All are Istinsim (finicky) about the cold.

Terumas ha'Deshen (66): 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.

Maharil (166): When it is so cold that I am in pain, I warm myself [through a fire made by Nochrim]. Maharam permits, but says that people are stringent [therefore he was stringent]. Nowadays, no one is stringent. If one is stringent, this is haughtiness, like we say about one who observes two days of Yom Kipur. However, when it is not so cold, in my eyes, it is a great Isur for one who does not have children in pain. I do not say so publicly, for it is better that Yisraelim transgress b'Shogeg than b'Mezid. Likewise, Yisre'elim may not heat the room on Yom Tov of Pesach, for sometimes it is not so cold. This is like incense. The Torah forbids it, for not everyone does so. Only pampered people do so.

Tosfos (Shabbos 107a DH umi'Mai): If the Mishnah of a needle is like R. Shimon, just like he permits Melachah she'Einah Tzerichah l'Gufah due to pain, also for an abscess. If it is even like R. Yehudah, who normally forbids Davar she'Eino Miskaven, but here he permits due to pain, likewise R. Shimon permits Melachah she'Einah Tzerichah l'Gufah due to pain of an abscess.

Tosfos (Kesuvos 60a DH Gone'ach): If [the one moaning] were danger, we would permit even a proper Melachah! In Yevamos 114a, we say that for pain, we permit only on Yom Tov! R. Tam answered that there it discusses pain of hunger. Here we discuss illness close to danger. The Ri says that our Sugya disagrees.

Poskim

Shulchan Aruch (OC 276:5): In cold lands, a Nochri may make a bonfire for minors, and adults may warm themselves from it.

Mishnah Berurah (39): If it is made equally for adults and minors, the Shulchan Aruch (325:6) forbids.

Shulchan Aruch (ibid.): He may make a fire even for adults if it is very cold, for all are sick regarding the cold.

Beis Yosef (DH Kasuv): If they are not sick, at least they are pained. One who is moaning may suckle from an animal (Kesuvos 60a). Even though it is Mefarek k'Leacher Yad, Chachamim did not decree [for one in pain]. I would permit explicitly telling a Nochri to light a fire for Yisre'elim, and all the more so if he does so by himself for Yisre'elim to warm themselves.

Beis Yosef (DH uv'Hagahos): Maharam holds that letter of the law it is permitted, but where people are stringent, he would protest against slaves who heat the room. Where there is no such custom, he would warm himself by a fire that a Nochri made for a Yisrael.

Beis Yosef (DH v'Chen): The Ra'avad explains that even though we do needs of a Choleh without danger through Nochrim, one may not make a bonfire. In any case, in very cold lands, perhaps everyone is like one who let blood and became chilled.

Mishnah Berurah (41): Normally, adults are not pained unless it is very cold.

Gra: Hagahos Mordechai brought many proofs. One can challenge all of them, except for piercing an abscess and using a needle to remove a thorn.

Damesek Eliezer: The proofs are that due to pain, we permit a Yisrael to do Shevus. All the more so we permit Amirah l'Nochri! One can challenge all the other proofs. Perhaps the moaning person is sick, or he is permitted only on Yom Tov. One may move a thorn due to danger to the Rabim. We are lenient for the Kohen Gadol because Ein Shevus b'Mikdash, or Chachamim did not decree about one who does not intend.

Mishnah Berurah (42): One may not light on Shabbos after Minchah for the sake of Motza'ei Shabbos, until Tzeis ha'Kochavim.

Mishnah Berurah (43): One must prepare wicks for the Nochri beforehand. A Yisrael may not move them for him.

Shulchan Aruch (ibid.): People who permit even when it is not so cold do improperly.

Beis Yosef (DH v'Chasav): R. Yerucham said that in these lands which are not so cold, it is forbidden, for people are not sick regarding the cold. Surely, the same applies to the coldest lands if it is not so cold on Shabbos.

Magen Avraham (15): Even if the Nochri did by himself, one must protest. Perhaps people rely on the Gemara (some texts have an abbreviation that could refer to Hagahos Mordechai or Hagahos Maimoniyos - Friedman Shulchan Aruch, Hagahah 47), which permits Shevus in a situation of pain (328:33). However, in 307:5, the Shulchan Aruch is stringent, and permits only Shevus d'Shevus. In any case, one may not tell a Nochri to light or do Melachah on Shabbos. Perhaps since one hires the Nochri to do so every day in winter, even if he does so when it is not so cold, it is as if he did so by himself. One need not leave the house.

Kaf ha'Chayim (49): L'Chatchilah, one must protest [against the Nochri].

Mishnah Berurah (44): We do not protest against people who are lenient, for it is better that they be Shogeg than Mezid.

Kaf ha'Chayim (49): The Levush says that people are lenient even when it is not so cold, because they never tried sitting in the cold, so due to habit it is as if they are sick regarding this.

Mishnah Berurah (45): One may not tell him on Shabbos.

Kaf ha'Chayim (52): If the stove broke, one may not tell a Nochri to fix it. He can go to a neighbor. All these Heterim are only in Ashkenaz, where it gets very cold. Heaven forbid to be lenient elsewhere! One who is stringent about this to honor Shabbos will live a long life.