Often when conspiracy theorists are told there's no evidence for their claims, they'll respond that "there's no evidence that my claim is untrue", falling back on the argument from ignorance. A priori I would expect the burden of proof to fall on the person who is making a positive claim, one that requires action, or alternatively that most deviates from Occam's razor. Is that not an accepted principle of logic? If not, how can any argument be settled? In the JFK assassination, for example, one could never prove that there wasn't a shooter on the grassy knoll. The only thing that can be proven with certainty is that there was a shooter in the School Book Depository. From a mathematical standpoint, the set of things that one would need to prove did not happen is infinite.
More broadly, how do you refute the argument of ignorance?