When the enchanted permanent becomes tapped, Brink of Disaster’s ability triggers. That permanent will be destroyed when the ability resolves, even if Brink of Disaster has left the battlefield or is somehow enchanting a different permanent by then.

If the enchanted permanent is tapped as a cost to activate a mana ability, the mana ability resolves immediately, then Brink of Disaster’s ability goes on the stack.

Brink of Disaster may target and may enchant a permanent that’s already tapped. It won’t do anything until the enchanted permanent changes from being untapped to being tapped.

If the enchanted permanent is tapped as a cost to activate an ability that’s not a mana ability, Brink of Disaster’s ability will go on the stack on top of that activated ability. Brink of Disaster’s ability resolves first (destroying that permanent), then the permanent’s activated ability resolves.