Many of these don't exactly count as Chekhov's guns, but they have this in common with your Chekhov's guns: They seem like substantial unresolved things and I will be disappointed if the end of HPMOR leaves a lot of them unresolved:

• Prophecies about Harry and the end of the world (for some values of "end" and "world").
• How magic works (e.g., why you have to say "Wingardium Leviosa" to make things float; resolving this may be more or less equivalent to resolving spell creation).
• Harry's intention of defeating death, perhaps in some fashion that involves the Deathly Hallows.
• The list of locations discussed by Harry and Quirrell, which may or may not correspond to Horcrux hiding places or something.
• Harry's "power that the Dark Lord knows not"; probably not either Science or partial transfiguration, but unlikely to be "love" as in Rowling.
• What's wrong (and how genuinely) with Quirrell.
• The interaction between Harry's and Quirrell's magic (kinda the same as in Rowling? maybe, or maybe not).
• Harry's vow to do away with Azkaban and the use of Dementors to guard human beings.
• Harry's debt to Lucius Malfoy. (Or -- I forget -- did that get cancelled somehow when Hermione got killed?)
• What, if anything, Harry was doing after Hermione's death; e.g., is he carrying her transfigured corpse around or something?
• Harry's "father's rock" (just transfiguration practice? actually some powerful magical artefact in disguise? etc.)
• What really happened in Godric's Hollow when Harry was a baby.
• Exactly what Quirrell's plans really are. (On some plausible theories, closely related to what happened in Godric's Hollow.)