If Burn from Within doesn’t deal damage to the target creature (perhaps because that damage was prevented or X is 0), neither additional effect will apply. It won’t lose indestructible, and it won’t be exiled instead of dying that turn.

You can target a creature that doesn’t have indestructible with Burn from Within. It will still be exiled if it would die this turn.

The damaged creature will be exiled if it would die for any reason that turn, not just if it dies due to damage from Burn from Within.