If a permanent enters under an opponent's control with hexproof or indestructible after The Fire Nation Drill's first activated ability resolves, it won't lose that ability unless you activate The Fire Nation Drill's ability again. The same is true if an opponent's permanent gains indestructible or hexproof after The Fire Nation Drill's ability resolves.

Because damage remains marked on a creature until the damage is removed as the turn ends, damage previously dealt to a creature with indestructible may cause it to be destroyed if The Fire Nation Drill's first activated ability resolves during that turn.