There are a few reasons to bury parts of the human body, and we will have to examine each one to see whether it applies to the tooth of a live person.
There is a major question regarding whether the obligatory mitzva of kevura (burial) applies even to an individual limb of a deceased person.[1] (If the majority of the corpse is present, all agree that the mitzva applies to every part of it.) Some maintain that while there is a mitzva of burial for an individual part of the body, this applies only to a dead person. Since all parts of his body should be buried, it is a disgrace for part of his body to go unburied. In contrast, when a person continues living without a part of his body, burial might not be necessary for that part.[2] The Igrot Moshe[3] attempts to prove that there is an obligation of burial even for an individual limb or a sizable amount of flesh of a live person, based on the fact that the gemara[4] needs a pasuk to deduce that a kohen may not make himself tamei in order to bury a limb of his live father. However, it appears that the majority opinion is that there is no obligation of kevura for a live person’s limb.[5]
Even if the mitzva of burial does not apply, there are additional issues that may put limitations on what to do with a part of a body. A dismembered limb of even a live person is tamei (ritually impure).[6] As such, it is not proper to allow such a limb to be left out in the open, where a kohen might come in contact with it. Another issue that could create requirements for proper disposal of the body part is the prohibition to receive benefit from it.[7]
Let us now look into the matter of a tooth. The gemara[8] relates that R. Yochanan would walk around and show people a bone from his tenth son to have died. (Tosafot[9] explains that he did so to console others who were distraught over tragedy.) Commentators are troubled how R. Yochanan could have acted in this way, as he ostensibly should have buried the bone. Rashi[10] posits that it was a tiny bone, smaller than the size of a grain of barley, whose halachot are different than those for a larger bone. The Rashbam[11] says that the “bone” was actually a tooth, which is not tamei. Indeed, a mishna[12] states that neither hair, nor nails, nor teeth that are separated from the human body are tamei. Although not everyone explains that it was a tooth that R. Yochanan kept, we do not have an indication that there is any fundamental machloket on a tooth’s status regarding burial. It is not clear whether it is permitted to benefit from a tooth from a deceased person,[13] but one certainly may benefit from the tooth of a live person. Therefore, we are not aware of any halachic source or strong reason to place restrictions on what may be done with an extracted tooth.
There is a venerable source that does discuss what one should do with extracted or fallen teeth, but it is apparently not following halachic lines. The Chida[14] wrote to someone who was upset that his tooth, which he had been saving to have buried with him, was lost. The Chida told him that although the custom of saving a tooth for eventual burial is mentioned in Ma’avar Yabok, it is not clear that the author agreed with that practice, and it appears to contradict the story of R. Yochanan cited above. The Tzitz Eliezer[15] seems to dismiss the practice. It is also interesting that even this uncommon minhag does not require one to bury the tooth right away, as many require for limbs, but rather to specifically wait until the burial of the person himself. This actually seems to discount the potential halachic stringencies, as we posited above. Therefore, we see no reason to take any special measures in regard to a tooth that is removed or falls from a live person.[16]