The first episode of Game of Thrones' eighth and final season concluded with Jon Snow learning the truth of his identity. Thanks to Sam and Bran, Jon now knows that he's the only child of Rhaegar Targaryen and Lyanna Stark, which makes him the true heir to the Iron Throne (if you support the Targaryen dynasty, that is). His claim is even stronger than Daenerys', but does Jon Snow want the crown now?
Unless he suddenly gets a personality transplant, it seems unlikely that Jon would make a bid for the throne. He spent the entirety of Episode 1 refuting his own claim to kingship, reiterating over and over again that titles didn't matter in the face of protecting his people. He tells minuscule boss Lyanna Mormont that while he appreciated the honor of being made King in the North, he would always choose safeguarding his people over a crown. Later, he tells Sansa that he never wanted a crown at all. His mission is to stop the oncoming army of White Walkers from decimating everything in their path. Nothing else matters to him but that.
He also supports Daenerys wholeheartedly. In Season 7, he bent the knee to her and swore his allegiance, because he needed her as an ally and believed that she would be a good queen. Since his introduction, Jon has never been a character that courted glory. Instead, he does what he must.
But it's possible that Jon must take the throne. He's challenged several times in Episode 1 over his support of Dany and his decision to give up his title as King in the North. The other Northern lords balk at supporting him now that he's backing Dany, presumably because they were hoping to be an independent kingdom once again, no longer at the mercy of an outsider's whims. Sansa wonders if Jon is Team Dany because of his feelings for her, something he neither confirms nor denies.
And in the final scene between Sam and Jon, Sam seems to plant some serious doubts in Jon's mind. When Jon is reeling over the news of his birth parents, he insists that Dany is their queen. But Sam points out that Jon gave up his crown to save his people. Would Dany do the same? That question appears to have an impact on Jon, though he doesn't answer it. Dany has been intensely focused on accumulating power, always listing her titles and demanding allegiance before she will ally with someone. Which is more important to Daenerys, the people or the power?
Jon's resistance to ruling proves that he's not actually all that interested in being the king. But that doesn't mean he won't end up in the position anyway. If he thought it was a burden he must take on for the greater good, then I could see Jon following through on staking his claim as Rhaegar's son. Plus, Game of Thrones thrives on misdirects and irony. If Jon doesn't want to rule, then odds are he'll end up on that throne somehow.