In Siman 303, Maran discusses various items which may not be worn in the public domain on Shabbat, out of concern that one may remove it and carry it. This could happen because either he wants to show it off to someone else, or he is embarrassed and wants to hide it from someone else.
One example of an item that a person may be embarrassed by is a gold tooth. Therefore, the Shulhan Aruch prohibits walking in the public domain with such a tooth. On the other hand, a silver tooth does not present a problem because it does not stand out in the same way. Hacham Bension (Or Lesion 23:1) rules that nowadays even a gold tooth is not a problem. Today, such teeth do not embarrass people. Moreover, even if one would want to remove it, they are implanted in a permanent fashion.
He also rules that a bite plate is not a problem for the same reason. He reasons that it is no different from a Kami’ah (amulet) worn for health. Braces would certainly be permissible, because they are attached. Yalkut Yosef rules that dentures are not a problem to wear in the public domain. It is not likely that someone would remove them in public, and there is nothing embarrassing about them. Similarly, there is no problem to wear Invisalign.
SUMMARY
It is permitted to wear gold teeth, dentures, bite plates, braces and Invisalign in the public domain on Shabbat.
