Here's the policy I would favor. I'm open to being contradicted if you've had experiences on other sites (Theoretical Computer Science, Math, Physics, …) that show that this policy is not viable.
Questions like “Is this proof correct?” or “I'm sure this proof is wrong but what's the fallacy?” or similar variants should be closed as “not a real question”, because they are overly broad (or worse, rhetorical like “I challenge you to find a fallacy”). You can't expect a review of a proof of anything more than a few clearly written lines in a Stack Exchange answer. Favorite crank topics can be held to a particularly high standard.
Unclear problems should be closed as “not a real question” too (ambiguous, vague, incomplete). If someone who's reasonably versed in the domain of the question cannot determine the precise statement of the problem from the question, close. Again, favorite crank topics can be held to a particularly high standard.
On the other hands, questions like “Is [clearly stated problem] in P” are valid questions. I don't see any grounds to refuse them. Again, the more the topic attracts cranks, the more precision and clarity should be required. If that problem is a variation on some well-known NP-complete problem, the answer “here is why this problem is NP-complete, and it is a well-known open problem whether it is in P” is a perfectly acceptable answer. If the question calls for more than that, close as “not a real question” (overly broad).
Furthermore, we need to be careful not to encourage copycats. This means quickly closing “dodgy” questions, and subsequently deleting them unless salvaged. Also, borderline questions should be edited to deemphasize the crank relationship.
With regards to Variant of TSP in P?, I have voted to close as “not a real question”. I left the following comment:
You are claiming that some variation of a well-known NP-complete problem is in P. This is an extraordinary claim requiring extraordinary evidence. Some random article found on the web does not constitute extraordinary evidence. Therefore I am voting to close this question as overly broad; it is not our goal here to make broad advances to science in a single post.
Thanks to Artem Kaznatcheev for setting me straight in chat about what the question was really asking.