Where I live, removeable items like braces, jewellery, et cetera are considered ‘personal effects’ and will be handed over to next of kin when they claim the body for funeral arrangements. Integral things like fillings, artificial joints, etc. are generally left inside. There are cases where they have to be removed or will be left behind (eg. cremation), but they’re still considered human remains and have to be disposed of properly.

There are also cases where things can’t be cremated or left in the body. I’m thinking specifically about pacemakers powered by radioactive isotopes. Medical authorities will take charge of those.