It may be relevant to downvote such a question, but that wouldn't merely be because it is asking for proof of something that is false. Something more is needed -- for example, sometimes a user will post a string of questions that boil down to "here's a random conjecture I made for no particular reason without verifying even the single-digit instances; please prove it for me". I would downvote that -- and the fact that the conjecture is easily false would contribute to my feeling that the OP ought to research his questions better before asking. But that kind of cases is an exception.
In any case, closing questions just because the ask for something false to be proved would be wildly inappropriate. Except for trivial typos, such a question always indicates that the asker has some misconception -- either about what is true or about which assumption can reasonably be left unsaid, or sometimes about notation where the asker copies a formula wrong because he's not aware that some detail carries meaning. In each of those cases it should be our mission to disabuse the asker of that misconception. Sometimes a nudge in comments will be enough, but the primary tool for this should be explaining in an answer why the statement is false, and perhaps take a guess at what the asker really wanted to prove.
Closing the question would prevent explaining to the asker what is wrong with his claim. It doesn't help anyone just to see "this question is closed because what it asks for a proof of is false". Just asserting that it is false falls short of what I think should be our standard -- if we're confident enough that it is false to close the question, we should be able to produce an actual disproof.
If anything, a question that clearly points towards a concrete misunderstanding of the asker's is miles beyond the standard "here's my homework and I don't know how to start", which doesn't point towards anything but laziness.
Of course, once an answer explaining why the desired conclusion is wrong has been posted, the OP should not edit the question to "correct" the statement, but rather post a new question. But that is a different problem, and not one that would be solved by closing the original question without even telling the OP why his claim is false.