Gemara

(Rava): "Kol Davar Asher Yavo ba'Esh Ta'aviru ba'Esh v'Taher" - the Torah requires an additional Taharah after Kashering (a Keli acquired from a Nochri).

(Bar Kapara): "B'Mei Nidah Yischata" obligates immersing it in water that a Nidah could immerse in, i.e. a Mikveh of 40 Sa'im.

(Rav Nachman): The Parshah (that obligates Tevilah) discusses only Kelim (similar to those used with fire, i.e. Kelim) used for food.

R. Ya'akov: The Parshah discusses only metal Kelim.

(Rav Ashi): Since glass Kelim can be (melted and) made again if they break, they resemble metal Kelim, and they require immersion.

The Halachah is, if an earthenware Keli was plated with lead, it has the law of the plating (it must be immersed).

Nidah 46a (Rava): A girl may not do Mi'un (retroactively annul a marriage mid'Rabanan) or Chalitzah after 12 full years.

Rava is not in Safek whether she is an adult, for he taught that we need not check a girl above 12. She is Muchzekes to be an adult.

Normally, we assume that she is an adult. She may not do Chalitzah in a case when we see that she does not have hairs now. She may not do Mi'un, lest she brought hairs after 12 years, and they fell out.

According to the opinion that is not concerned lest hairs fell out, we can say that we did not check her. Rava's Chazakah is for Mi'un (to forbid it). She may not do Chalitzah until we see hairs.

(Rav Dimi of Neharde'a): The Halachah is, we are concerned lest hairs fell out (regarding Mi'un) only if she had Bi'ah after 12 years, for then it is a Safek mid'Oraisa). If not, she may do Mi'un (she is married only mid'Rabanan).

Rishonim

Tosfos (Pesachim 4b DH Hemuhu): We believe a minor about Bedikas Chametz, for he can do so himself. He cannot change Techum Shabbos, so he is believed about it only after he matures, even though it is mid'Rabanan.

Terumas ha'Deshen (257, brought in Beis Yosef YD 120 DH Kasuv bi'Terumas): The Torah requires Tevilah of metal Kelim, so one may not believe a minor to say that he did so. Even though he is able to, and we believe minors about something without exertion, this is only for mid'Rabanan matters. Tosfos (Pesachim 4b) says that women are slaves are believed about mid'Oraisa matters in their control. He did not say so about minors. Hagahos Ashri (Chulin 2:8) says that a Nidah must have intent while immersing for her husband, even though this is Chulin. Tosfos (Chulin 12b DH v'Tibo'i) says that with an adult supervising and instructing him, a minor can have intent even to write a Get. It seems that for Tevilas Kelim, one can teach a minor to intend for Taharah even if the adult will not be there at the time of Tevilah. Tevilas Klei Nochri is not for Taharah. Perhaps it does not require any intent!

Poskim

Shulchan Aruch (YD 120:14): We do not trust a minor about Tevilas Kelim

Taz (16 and Gra 36): This is because it is mid'Oraisa.

Mishbetzos Zahav (OC 451:6): This is even for glass Kelim, for which Tevilah is mid'Rabanan, since there is Chezkas (a status quo of) Isur. However, for glass Kelim, if a boy is 13 years old, we need not check if he brought hairs. We must check him for metal Kelim.

Pischei Teshuvah (14): Perhaps we need not check even for metal Kelim. This is more lenient than other Isurim mid'Oraisa, for if one transgressed and used a Keli without Tevilah, it does not forbid the contents.

R. Akiva Eiger (on Taz 16): All agree that Tevilah of glass Kelim is mid'Rabanan, so a minor should be believed. However, Tosfos (Eruvin 31b DH Kan) says that a minor is believed about mid'Rabanan laws that apply to him, but not about a Shlichus for someone else.

Aruch ha'Shulchan (13): We are more lenient about Tevilas Kelim because it is not an Isur. It is only a Mitzvah.

Tzitz Eliezer (13:75:2): Ha'Gaon R. Y.A. mi'Kovno says that the Acharonim agree that one may rely on a minor for Tevilah of glass Kelim, even before 13 years. One may not rely on him for metal, even if he is above 13 if he was not checked for hairs. We do not rely on Chazakah d'Rava for mid'Oraisa laws. I do not accept Pischei Teshuvah's distinction. However, one could be lenient because Rishonim argue about whether Tevilah of metal Kelim is mid'Oraisa. The Pri Chodosh says that the Shulchan Aruch rules that it is mid'Oraisa, but there is no necessity to say so. Yeshu'os Yakov says that most Poskim say that it is mid'Rabanan. I say that it is a Safek. The Shach (Hilchos Sefek Sefekah 16) says that (even) if one Safek of a Sefek Sefekah is a Safek Halachah, this is a Sefek Sefekah. Here, perhaps Tevilah of metal Kelim is mid'Rabanan (and we can rely on Chazakah d'Rava), and even if it is mid'Oraisa, perhaps he brought two hairs. It would seem like a Sefek Sefekah against a Chazakah. However, Pischei Teshuvah said that this is not Chezkas Isur, for the contents do not become forbidden. Also, the Pri Megadim (on Shach YD 103) says that we rely on a Sefek Sefekah against a Chazakah, even if one Safek is a Safek Halachah. One may rely on what I wrote, for the Maharit (CM 41) says that Chazakah d'Rava is a strong Chazakah.

Yabi'a Omer (7 YD 9:4): Amudei Ohr (59:8) disagrees with the Pischei Teshuvah. Even though the Keli does not forbid the food, the Torah obligates Tevilah, so one may not rely on a youth without hairs. Perhaps one may be lenient because it is a Sefek Sefekah. However, the Mishbetzos Zahav says that one may not rely on minors even for what is mid'Rabanan, for there is Chezkas Isur, like the Rema (127:3). If so, the Sefek Sefekah forbids, for perhaps a minor is not believed even for mid'Rabanan, and if he is, perhaps Tevilah of metal Kelim is mid'Oraisa. However, since the Safek that he brought hairs is even, we may join the Safek that a minor is believed about metal Kelim, even though it leans to forbid, and this is like a majority to permit. Ein Yitzchak (2 EH 16 3-4) and Oneg Yom Tov (45) prove this from Tosfos Nidah 45a; it seems that Tosfos Kesuvos 9a disagrees. R. Akiva Eiger and the Gra (127:32) explain that the Terumas ha'Deshen believes minors about glass Kelim because it is not considered Chezkas Isur, because the minor can immerse them.

Rema: If a minor immersed a Keli in front of an adult, it is a Tevilah.

Bach (DH ha'Koneh): R. Menachem says that if a Nochri's Keli fell into water, or one washed his hands without intent and before diverting his mind, he decided to eat, he need not immerse or wash again. This is like the Rambam, who holds that Tevilas Nidah does not need intent. We rule that it needs intent. The Shulchan Aruch (Sa'if 15) rules that if a Nochri immersed a Keli, this was Tevilah. This is from Teshuvas ha'Rashba, but he holds that Tevilas Nidah does not need intent. The Halachah does not follow him. One should immerse it again without a Berachah.

Rebuttal (Shach 28, Taz 16,17): The Rashba requires intent for a Nidah (Beis Yosef Sof Siman 198), and for Netilas Yadayim! Even so, he is lenient about Tevilas Kelim. The Rema does not require the adult to teach the child. He holds like the latter answer of Terumas ha'Deshen, that no intent is required.

Gra (37,38): We hold that no intent is needed for Chulin (Chulin 31a). The Bach is correct. (The Acharonim say that one cannot prove how the Rashba holds.)

Taz (17): One may use a minor l'Chatchilah for Tevilas Kelim, but one may not use a Nochri l'Chatchilah, for one cannot bless on his Tevilah. If one blesses on a Keli that he himself immerses, he may tell a Nochri to help immerse other Kelim.

Rema (127:3, b'Sof): If a minor testifies to be lenient about an Isur mid'Rabanan without Chezkas Isur, e.g. Bedikas Chametz, Chachamim trusted him about this. If there is Chezkas Isur, he is not believed at all.

Gra (32): If something is in a minor's control, he is believed even if it is mid'Oraisa, unless there is a Safek (i.e. a reason to be lenient). He is believed about a mid'Rabanan Isur only if it is in his control; this is even if there is a Safek.

Note: A Hagahah alludes to the Gra 120:36, which seems to say that a minor is never believed about a mid'Oraisa Isur.