Do special abilities still function after death?

I know that, for example, the paladin’s class feature Aura of Courage specifies that it doesn’t function if the paladin is dead. But what about passive abilities granted by race?

For example, can you take out a beholder’s antimagic eye (or lift the whole dead creature) and point it at enemy mages to prevent them from casting spells (supernatural ability)? Can you glue a hellcat’s severed limb (or full corpse) to a doorway to create an invisible barrier (extraordinary ability)? Can you cook meat from a red dragon (or roast it whole) (extraordinary ability but with a possibly different source)?