The broadest complaint I have of this nature is when one claims it's somehow deficient of me to demand experimental evidence proving a claimed definitive beneficial effect. It's worse when there is conclusive experimental evidence that something does not work yet I'm confronted with someone who believes it does. Case in point: homeopathy.
Worse still, IMHO - people who claim to believe in something for which they can provide no evidence, and yet categorically state that nothing will ever change their minds.
I specifically remember this one from a discussion about the alleged harmful effects of wi-fi radiation. Somebody (whose name I forget, so for the sake of argument I'll go with "the Idiot") claimed that a neighbour's wi-fi was stopping him sleeping, giving him headaches or such - and that nothing would ever convince him otherwise.
The only possible conclusions from this are that either:
a) the Idiot is in possession of some solid, incontrovertible piece of evidence that he's correct, but refuses to share it for some reason, or
b) he doesn't base his opinions on objective evidence at all, and therefore any attempt to inform, educate or placate him is a complete waste of time.
I'm inclined to believe it's (b), though obviously I'm prepared to believe (a) if presented with the evidence.