Gemara

(Mishnah): If a handle of a Kli is too long and is destined to be cut, when immersing the Kli, one must immerse as much of the handle as he will keep;

R. Yehudah says, the entire handle must be immersed.

Rishonim

Rosh (Hilchos Mikva'os 26): A Tosefta says that tight rings and bracelets are a Chatzitzah. What is the reason? It is on the minority, and she is not particular about it! The Ra'avad says that it is because she is particular about it to remove it when kneading. Since at some time she is particular about it, even though she is not particular about it at the time of Tevilah, it is a Chatzitzah.

Rosh (30): The Halachah follows the first Tana. If a handle of a Kli is too long, one must immerse as much of the handle as he will keep. A Tosefta explains why the place it will be cut is not a Chatzitzah. We are not concerned for Beis ha'Setarim of Kelim.

R. Shimshon (Mikva'os 10:8 DH bi'Zman): In a Tosefta, R. Meir says that an arrow is not a Chatzitzah, and Chachamim disagree. They discuss metal. It is not a Chatzitzah, for water enters. It does not cling to the flesh as much. Alternatively, it is dangerous to remove metal. If one is not particular about something, it is not a Chatzitzah.

Poskim

Shulchan Aruch (YD 198:1): A woman must immerse without any Chatzitzah. If sometimes people are particular about it, it is a Chatzitzah even if she is not particular about it now, or even if she is never particular about it.

Shulchan Aruch (8): Mascara in the eye it is not a Chatzitzah. If it is outside the eye, it is a Chatzitzah.

Shach (14): If she paints her eyes for Refu'ah, it is a Chatzitzah. If it is for appearance, it is not a Chatzitzah.

Shulchan Aruch (24): A woman must clean her teeth (before Tevilah) so there will be no Chatzitzah. If she immersed and found something stuck to them, the Tevilah was invalid.

Hagahos Yad Sha'ul: Doctors made a lead filling for a cavity, and said that they will remove it after three months. Surely, it is not a Chatzitzah. One is not particular about something for Refu'ah! Women would not be adamant about holes in their teeth. The Rema (Sa'if 6) says that one is not Makpid (particular) about anything that entails danger. R. Shimshon says that metal is not a Chatzitzah because there is danger.

Chachmas Adam (Beis ha'Nashim 21, Binas Adam 12): Some women with toothaches due to cavities have doctors fill the holes with wax or lead. Regarding Chatzitzah, I compare this to dye on (the hands or face of) women or on dyers (Saif 17). It is not a Chatzitzah. The Rashba says 'it is not a Chatzitzah for it is an adornment for them. This is even if it is on all the hair. Even though the majority is a Chatzitzah even if one is not particular, here she wants it. It is like the hair itself. Also, there is no real substance in the hair or hands. It is a mere appearance. This is unlike writing. Ink has substance; it is a Chazakah. All women dye (their head or hands).' The Rosh and Ramban (Teshuvah 124) say similarly. The Rashba's first reason is primary. Even if there is substance of the dye, it is not a Chatzitzah. This was taught with butchers, who have actual blood on their hands (and it is not a Chatzitzah). Also, the Rashba needed to say that it is only an appearance, for when it covers the majority, a substance would be a Chatzitzah even if she is not Makpid. The Shach says that mascara for Refu'ah is a Chatzitzah, but not if she intends for adornment. For adornment, or one's profession, even the substance is not a Chatzitzah. For Refu'ah, e.g. a bandage, she does not want it there, just she cannot remove it. Therefore, it is a Chatzitzah. Therefore, we needed to say that danger permits Marlakan (a cure that causes hair to stick together). If so, a false tooth is not a Chatzitzah, for it is for adornment. A filling is a Chatzitzah. It is for Refu'ah; she is particular about it. The Rambam forbids going out on Shabbos with a false tooth. He permits a silver cover for a black tooth, for then it is not recognized (that it is different from her teeth). Whatever is a Chatzitzah, one may not go out with it on Shabbos. Really, all false teeth should be permitted, just if it is not of silver we decree lest others mock her and she will remove it. This implies that a filling, which is like a silver cover, is permitted. However, this is wrong. The silver cover is for adornment, therefore it is permitted. A filling is for Refu'ah. She has it only due to the pain.

Igros Moshe (YD 1:97): Sometimes, they fill a cavity with a filling for a month or more, and afterwards they replace it with a permanent filling. For the specified time, she wants it to be there. The questioner said that this is called that she is not particular to remove it, even though she is adamant to remove it after the time. This is unlike YD 198:1, which is based on the Rosh citing the Ra'avad. When she is not kneading, she is particular about it not to remove her ring, for it an adornment! Also here, the filling should be a Chatzitzah, since she intends to remove it afterwards. However, Zichron Yosef (YD 10) distinguishes. Hakpadah only after a long time is not considered Hakpadah now. He does not distinguish between after a long time, e.g. a year, and a short time, e.g. less than a month. There is no source for such a distinction! Rather, there is no fixed time for kneading. It can occur any day or even at night. Therefore, it is as if she is always particular. If she removes it only after a set period of time, until then surely she will not remove it. She is not particular until then. It does not matter if the fixed time is after a long or short period. If so, she is not particular about a temporary filling.

Igros Moshe (DH Aval): The Noda bi'Yehudah and Chasam Sofer (cited in Pischei Teshuvah 16) agree that if a woman must remove an internal ring at the time of Hefsek Taharah, this is considered Hakpadah, even though it has a fixed time, and she is adamant that it be there until then. This is unlike Zichron Yosef. However, we can say that Hefsek Taharah has no fixed time. Even if she has a regular menstrual cycle, perhaps it will change, and right after immersing she will see blood or a stain and need another Hefsek Taharah. They could agree that a temporary filling it is not considered Hakpadah.

Igros Moshe (DH Ach): However, dangling limbs or flesh are a Chatzitzah (198:22). The Beis Yosef explains that even though the place was never exposed, since it ceases to nourish from the body, they are not considered part of the body. They are destined to be cut, so it is as if they were cut, and water must be able to come there. The limb prevents this. He gives another Perush that differs only about what is not exposed, but it could agree that what is exposed is Chotzetz. If so, water must be able to reach the place where the thorn entered! We must say that if the thorn would be removed, it would heal and the area would be covered (by a scab, and then skin). Therefore, it is considered sealed even now. Alternatively, the thorn is small, and even if it were removed, water would not enter, therefore it is not a Chatzitzah. R. Akiva Eiger (60) says so, but the first answer is correct. A hole in a tooth was exposed, and it would not cover itself, and water could have entered. However, it is a Chidush that water must be able to enter the mouth. Perhaps we are not so stringent about Beis ha'Setarim in the mouth as in other places. R. Shimshon and the Tosfos Yom Tov support this. However, it is hard to rely on them to make a new Heter.

Igros Moshe (Anaf 2,4): We can say that a Chatzitzah without Hakpadah is not a Chatzitzah (1) because it is Batel to the body, or (2) because Chatzitzah is only with Hakpadah. The questioner wanted to be lenient about a filling for a third reason. A filling in a hole in a tooth is like a thorn in the flesh. If it is low, it is considered not seen. The Shach (198:16) is stringent when it is even with the flesh, due to a Safek. The Rosh says that we are concerned lest some stick above the flesh. This applies to flesh, which is soft, but not to teeth, which are hard and do not bend. I say that the Safek is when the thorn is sunken in, but if it is even with the flesh, it is surely a Chatzitzah.

Igros Moshe (6 DH v'Hinei): If a bad filling was put in before the tooth healed, even if it does not hurt but one cannot chew with it, surely it will be removed and another filling will be put in. The first two reasons to permit do not apply. This is like abnormal handles of Kelim. They are a Chatzitzah. It is hard to rely on the third reason without a proof, for it is a Chidush.

Igros Moshe (DH Nimtza): Based on the third reason, letter of the law there is room to be stringent about a temporary filling. There is a small concern based on the first reason, but no reason at all according to the second reason. If a bad filling does not cover any of the tooth, it seems that it is not a Chatzitzah, but it is hard to be lenient in practice. Surely one may be lenient about a good filling, even if it is temporary.

Minchas Yitzchak (5:111 DH uv'Zeh): The Poskim agree that a permanent filling is not a Chatzitzah. Therefore, if it will be permanent, just once in a while they must switch the material, it is like a permanent filling, and all the more so if the temporary filling is for an unspecified amount of time. However, one must ensure that it does not cover the tooth more than the permanent filling will. If it does, the Heter does not apply. Chelkas Yakov (1:137) says that the temporary filling should be a little sunken, so it will be enveloped. This is proper.

Shevet ha'Levi (2:98): If a person can remove and insert a false tooth without a professional, surely it is a Chatzitzah. A permanent filling is not a Chatzitzah. If one is particular not to remove a temporary filling prematurely lest he expose the nerve, surely it is not a Chatzitzah. Once it is time to remove it, and one could get sick if it is not removed in time, it seems that it is a Chatzitzah. One must be careful about bridges (braces that hold artificial teeth in place), for often food accumulates there and it is hard to reach there (to clean it). If one has braces around shaky natural teeth, and they can be removed, and one removes them occasionally, it is a Chatzitzah. One may be lenient if it is impossible to remove them.