Jon is clearly the noble but reluctant leader who gets thrust into positions of leadership instead of seeking them out. He's clearly not terribly enthused about being King in the North, so unless Dany abruptly decides to abdicate her claim in favor of his he's not going to challenge her for the throne. While the succession laws are important, they certainly aren't set in stone since a bastard was elected King in the North in place of a true-born daughter. Seeing as how Dany and Jon are now working closely together, if the truth of his parentage does come up I can see him wanting it kept quiet to keep the North stable (who knows how they'd react to finding out they elected a Targaryen as King in the North) and to keep from weakening Dany's claim.
Seems very likely that the truth of his parentage will become generally known at some point, I'm guessing Bran will just casually blurt it out in front of the worst possible witnesses, or Littlefinger will somehow survive whatever retarded plot he's currently fomenting and overhear Sam telling the truth to Sansa.