Unlike some other Pokémon, Unown don't have any special abilities or events triggered when they faint. They simply act like any other Pokémon that faints:
* They are unable to battle until healed.
* They are sent back to the PC if you have no other Pokémon left in your party.
While some fans speculate about Unown having hidden potential or events related to their fainting, there's no concrete evidence or gameplay mechanic to support these ideas in Pokémon HeartGold.