Fainting in both games and anime are just portrayed like 'they couldn't fight anymore due to exhaustion!' or 'they simply don't have the will to continue the battle and thus stay away for some time', and some us us advanced challengers interpret fainting to be equivalent of that 'perma-death'. I still have my regrets and I couldn't get over it if I were to recap the past events.
Though there's still something that I'm not sure about. If a Pokémon was afflicted by a major status ailment like burn and it fainted, and then revived again, the status itself will be gone. Technically the burn will still stay there, and this is somehow legit then this would also encourages players to actually bring status curing items, though I would assume this is to simply balance out the game.