Quick question: is this even anything? A barbarian diagnosed with Congenital Insensitivity to Pain (CIP).
He'd never feel physical pain of any kind, but that makes him not realize he's being eaten alive by acid/burning to death.
The player would also never know his health bar. That would be maintained by the DM.
I'm writing this at 1 in the morning and thought it would be a neat idea for a character.