Gemara

(R. Ami): From the lips and inside is considered an internal wound (for which one may desecrate Shabbos).

Question (R. Elazar): What is the law of a wound on the Kachi or teeth?

Since they are hard, this is like an external wound. Or, since they are internal, perhaps it is considered an internal wound!

Answer #1 (Abaye - Mishnah): If one's teeth hurt, he may not be Yigma (put in his mouth and spit out; Rava - swallow) vinegar (on Shabbos).

Inference: If he is in great pain, this is permitted (for it is considered an internal wound)!

Rejection: The Tana taught 'hurt', and this includes a great pain.

Shabbos 111a (Mishnah): If one's teeth hurt, he may not Yigma vinegar. He may eat vegetables dipped in vinegar. If he is healed, he is healed.

Contradiction (Beraisa): He may not Yigma and spit out, but he may ingest and swallow.

Resolution #1 (Abaye): Also the Mishnah forbids (only) to put in his mouth and spit out.

Resolution #2 (Rava): It forbids even to swallow, but only after eating (it is clearly to heal). The Beraisa permits before eating (this is a normal alternative to dipping in vinegar).

Question: Migo (since) we permit before eating, we should permit also after eating (even though it is clearly to heal). Rava himself applies Migo!

Answer: Rava retracted from his explanation of our Mishnah.

Rishonim

Rambam (Hilchos Shabbos 2:5): One may desecrate Shabbos for any internal wound, i.e. from the lips and inside, including the teeth. This is only if an internal limb was ruined due to a blow, blister, etc. Pains are not called a blow.

Ran (Shabbos 39b DH umi'Ha): If a person has a mere pain, but he is strong and walks like a healthy person, he may not do anything that is evidently done for Refu'ah. We learned that if one's teeth hurt, he may not Yigma vinegar, even if he has a great pain, since it is not an illness. When there is mortal danger, adult Yisre'elim do all his needs. If there is no mortal danger, just danger to a limb, Nochrim do his needs that require a Melachah mid'Oraisa, and even Yisre'elim may transgress Shevus (mid'Rabanan laws) for his needs. If there is no danger to a limb, just mere illness, Nochrim may do even Melachos mid'Oraisa for his needs, but Yisre'elim may not transgress even Shevus for him. If there is mere pain, e.g. a toothache, there are no leniencies. Further, even things permitted to a healthy person are forbidden if it is evidently done for Refu'ah. The Ramban says so in Toras ha'Adam.

Question (Beis Yosef OC 328 DH Aval): The Ran rules like the rejection. This is astounding, for it was a mere Dichuy. We are lenient about a mortal Safek! Perhaps the Ran does not rule like this; he just explains the rejection, but he admits that it is a Safek and we are lenient.

Magen Avraham (328:2): The Ran (40a DH Min) explicitly says that the Safek was not resolved (so we are lenient). On 39a he merely proved from the rejection that a mere pain does not permit telling a Nochri to do what Yisre'elim may not do Shabbos.

Rosh (2:9): The question about Kachi and teeth was not resolved. We are lenient about a mortal Safek, so one may be Mechalel Shabbos for it.

Tosfos (28a DH Kachi): Rashi says that Kachi are gums. This is difficult, for the Gemara says that they are hard! Another Gemara discusses Kachi that fell out! Rather, they are big teeth.

Poskim

Shulchan Aruch (OC 328:3): One may desecrate Shabbos for any internal wound, i.e. from the lips and inside, including the teeth. This is only if an internal limb was ruined due to a blow, blister, etc. Pains are not called a blow.

Source: The Ran (Shabbos 40a DH Makah) says so in the name of R. Yonah and the Rashba, also brought in Beis Yosef DH v'Chasav.

Mishnah Berurah (7): All the more so, a wound in the gums is called an internal wound.

Mishnah Berurah (8): This applies if the entire body is sick due to the tooth, even if he need not lie down due to it.

Kaf ha'Chayim (11): Even a Yisrael may do a Melachah mid'Oraisa for a Stam tooth wound.

Kaf ha'Chayim (17): If the entire body is sick due to the tooth, one may tell a Nochri to do a Melachah mid'Oraisa. If not, one may tell him to do Shevus.

Rema: However, if one's tooth hurts and he wants to extract it, he tells a Nochri to extract it.

Beis Yosef (DH Kasuv b'Orchos): Orchos Chayim (Shabbos 334) says so, for when there is no danger, one tells a Nochri to do his needs.

Gra (6): The Gemara said that a great pain is different.

Bach (Sof 4): The Maharshal says that there is no proof to permit extracting a tooth through a Nochri dentist on Shabbos due to mortal danger. Chilul Shabbos mid'Oraisa is permitted only for giving medicines that are Vadai cures, e.g. to grind up ingredients. Extracting a tooth is also mortal danger (perhaps it will kill him)! However, the Beis Yosef says that Orchos Chayim permits, and the Rema rules like this.

Taz (1): The Yisrael helps him to extract it! Why is this different than a Nikaf (one whose sideburns are cut), who is liable (Makos 20b)? The Mefaresh (Rivan DH Mesaye'a) says that he presents his hairs to the barber. Here also, he presents his tooth to be extracted! Beitzah 22a says oppositely. Ameimar permits to have one's eye painted by a Nochri on Shabbos. Rav Ashi disagreed, for he helps, i.e. he opens and closes it. We answer that help is insignificant, i.e. in such a case when he does not do Melachah and merely helps a little, and even without this the Melachah would be done. We say similarly in Shabbos 93a. If so, why is a Nikaf different? One must do an action to be lashed! I did not see Tosfos or Poskim distinguish. It seems that the Halachah follows both of these. It is also a contradiction in Rav Ashi. Tosfos asked a contradiction only in Rav Zvid. We must say that Ameimar discusses one who does no action, for his eyes are already open. The Nikaf presented his hair. This is like starting the Melachah, and it is significant. The Tur says that he tilts himself towards the barber. If so, the same applies to one who opens his mouth and does an act before the Nochri extracts his tooth! Normally, one's mouth is closed, and he opens it for eating or other matters. If so, he is like a Nikaf, and it is forbidden. My distinction shows why if a woman who needs to immerse on Yom Tov Rishon forgot to cut her nails before Yom Tov, she may not tell a Nochris to cut them, for she must stick out her hand for this. This is like a Nikaf.

Rebuttal (Nekudas ha'Kesef YD 198:21): Most say that cutting nails on Shabbos is only mid'Rabanan even with a Keli. All agree that it is only mid'Rabanan to cut another's nails by hand. The RIvan (Makos 20b) says that "Lo Sakifu" connotes that one may not allow himself to be Nikaf. Alternatively, the plural obligates the Makif and Nikaf. A Nikaf is lashed only if he did an action to help. Regarding other Aveiros, help is insignificant. For the sake of a Mitzvah, one may tell a Nochri to do Shevus.

Mishnah Berurah (11): Even though he must open his mouth, many Acharonim hold like the Shulchan Aruch, that help is insignificant.

Magen Avraham (3): We must say that his entire body is sick due to the tooth, like the Rema says in Sa'if 17. Extracting a tooth is a Melachah mid'Oraisa, i.e. wounding for the sake of Refu'ah.

Bi'ur Halachah (DH u'Mitzta'er): According to the opinion that exempt for a destructive wound, this is not mid'Oraisa. He does not need the blood. Even if it is inevitable, one is not liable for this.

Kaf ha'Chayim (18): According to the opinion that exempts Melachah she'Ein Tzerichah l'Gufah (it is not needed for the same purpose as in the Mishkan), extracting a tooth is not a Melachah mid'Oraisa, so one may tell a Nochri to do it even if the entire body is not sick.

Shulchan Aruch (32): If one's teeth hurt, he may not put vinegar in his mouth and spit it out, but he may drink and swallow it or dip food in vinegar (and eat it) like normal.

Taz (24): This refers to a mere discomfort. If it is a great pain and the entire body is weakened, all Refu'os are permitted, like in Sa'if 3.

Mishnah Berurah (100): Since it is a mere discomfort, he may not tell a Nochri to do even Shevus. If it is a great pain and the entire body is weakened, he may tell him to do even a Melachah mid'Oraisa.

Mishnah Berurah (101): If he delays it in his mouth, it is evidently done for Refu'ah, so it is forbidden even if he later swallows it.

Mishnah Berurah (102): Whiskey is like vinegar. If one is in great pain, whiskey is permitted if he later swallows it.

Kaf ha'Chayim (193): One may swallow vinegar even if all know that he finished eating.