A. C. Grayling remarks in his short essay 'Proving a Negative' (in Thinking of Answers, pp.32-34) that "[t]he claim that negatives cannot be proved is beloved of theists who resist the assaults of sceptics by asserting that the non-existence of God cannot be proved." (p.32) It is indeed not uncommon for a theist to challenge an atheist with the question 'Can you prove that God doesn't exist?', where the rationale for posing the question is that it is impossible to prove a negative. It is, however, a little strange that Grayling does not mention also the fact that some atheists (and agnostics) like to invoke this very same principle of folk logic when they argue against theism: they would insist that the burden of proof lies with the theist because it is impossible to prove a negative but possible, and sometimes easy, to prove a positive.
In any case, it is puzzling why there are so many people who accept the principle that it is impossible to prove a negative, for the principle seems obviously false. If 'a negative' means simply 'a negative proposition', then anyone who accepts the law of non-contradiction (i.e. '~(p & ~p)') has to accept that there is at least one negative that can be proved: the law of non-contradiction is a negative proposition and it can be proved in classical logic. Relatedly, propositions of the form "'p' is not false" are negatives in this sense. Assuming bivalence, proving that 'p' is not false is equivalent to proving that 'p' is true --- unless propositions of the form "'p' is true" cannot be proved, it is possible to prove a negative.
Perhaps for those who accept the principle in question 'a negative' here means 'a negative existential proposition' (i.e. a proposition of the form 'There is no x' or 'x does not exist'). Grayling does not tackle the question why people accept the principle in question, but he does give a simple counterexample: "consider how you prove the absence of pennies in a piggy-bank. You break it open and look inside: it is empty." (p.34) I think most people would agree that doing so proves the negative 'There are no pennies in the piggy-bank'. So, even when 'a negative' means 'a negative existential proposition', it is still clear that a negative can be proved.
Perhaps the universe of discourse is supposed to be unrestricted when the principle in question is applied. It is easy to prove that there are no pennies in the piggy-bank, but, it may be claimed, it is impossible to prove that there are no unicorns, period (though it can be proved rather easily that there are no unicorns in your closet). There are two problems here. First, if the difficulty of proving that there are no unicorns anywhere in the universe is due to our technological limitations, then those who accept the principle in question have to clarify what they mean by 'impossible' when they claim that it is impossible to prove a negative. It is at least not a metaphysical or logical impossibility. Second, it is not clear that the universe of discourse matters here. It may be easy to prove that there are no unicorns in your closet, but it is difficult to prove that there are no invisible unicorns in your closet. And it seems impossible to prove that there are no unicorns of the following kind in your closet: unicorns that are undetectable unless they allow you to sense their presence.
If it is impossible to prove that God does not exist, it may be because God is undetectable unless God allows us to sense her presence, rather than because 'God does not exist' is an unrestricted negative existential proposition. It remains a puzzle to me why there are so many people who accept that it is impossible to prove a negative.