If there is no empirical evidence either way about a belief, how would one go about destroying it?

One way you could go about destroying a belief like that is to use Ockham's Razor: sure, it's possible that all kinds of unfalsifiable beliefs are true, but why should you waste time in believing any of them, if they have no effect on anything ?

However, if the believer has some subjective evidence for the belief -- f.ex., if he personally experienced the gods talking to him -- then this attack cannot work. In this case, would you still say that his belief is "indestructible" ?