Consider the demand for empirical evidence -- or the question (often rhetorical): "What is your empirical evidence for that claim?" -- where empirical evidence is evidence based directly on sensory experience (or something along those lines). Often the demand for empirical evidence is made (or requested) without the modifier "empirical" -- but it is assumed or understood that this is the sort of evidence being demanded (or requested).
Often empirical evidence is just the sort of evidence one needs in order to answer a question or settle an issue. For instance, if you want to know how many chairs are in the room or whether or not any trees are planted in the courtyard, empirical evidence is just the sort of evidence that is most appropriate. However, is that the only sort of evidence that is acceptable or legitimate? What sort of empirical evidence could settle the question of whether or not 2 is necessarily an even number? What kind of empirical evidence could refute (or establish) whether it is necessarily true (or not) that only nothing comes from nothing? Not even quantum indeterminacy or particles arising from minimal energy states could do that.
In any case, this sort of epistemological demand sometimes (perhaps even often) has as a background assumption that the only legitimate appeal to evidence is the appeal to empirical evidence. However, such a demand is self-defeating. This assumption has no empirical support itself. Further, an appeal to the success of science will not help here, for the most that this can show is that certain sorts of issues are best investigated by empirical (or scientific) means. In other words, there is no good empirical evidence that the only kind of genuine or real or legitimate evidence that one can have is empirical evidence. So, if the only grounds that one can have for rationally believing something is empirical evidence, then (by its own standard) no one can rationally believe the claim that only empirical evidence is legitimate evidence.
Nevertheless, the demand for empirical evidence as the only legitimate evidence is an extraordinarily pervasive demand on internet discussions -- but that doesn't make it any less self-defeating as a demand (or as a question or as an assumption). It is such a pervasive mistake that I think that it deserves its own name. To that end, I suggest the following:
"The Empiricist Fallacy"
Of course, I'm open to hearing the thoughtful, polite, and well-articulated considerations of others on this issue.