Ever since Khal Drogo died, Daenerys has rarely allowed genuine feelings to mix with her desire to rule, which is buoyed by her unwavering faith in her rightful claim to the Throne. Jon's revelation shatters the one certainty in her life. She reacts initially in almost petulant disbelief. (Her argument's not a bad one: So your brother and best friend told you this? Please.) But as Jon insists that it's true, something changes on her face. Once again, her look is hard to parse, but it seems like some combination of anger, love, and disillusionment. This, to Dany, is her world crumbling.
Per Emilia Clarke, Dany's not bothered by the fact that she's been getting it on with her nephew, given that incest is par for the course in the Targaryen family. Instead, it's Jon's claim that's shaking her, she told Entertainment Weekly. "This is my whole existence," she said. “Since birth! Dany literally was brought into this world going: RUN! These f—ers [in Westeros] have f—ed everything up. Now it’s, ‘You’re our only hope.’ There’s so much she’s taken on in her duty in life to rectify. There’s so much she’s seen and witnessed and been through and lost and suffered and hurt to get here … and Jon doesn’t even want it!”
For now, Jon and Dany have to go fight the White Walkers, but presuming they both survive, it doesn't seem like this conflict is going away any time soon. (By the way, there's also a deeply uncomfortable Jon/Sansa/Daenerys love triangle theory out there that Joanna Robinson wrote about on Vanity Fair, because you just can't squeeze enough incest into the final Thrones season.)
There is an easy, if unlikely, solution. Jon doesn't want the Throne so he gives it to Daenerys, who allows Sansa to retain control of the North, seeing a kindred spirit in the young woman who did everything she could to reclaim her home from her rapist. But that would require Dany to basically reshape her worldview. The threat of Dany's own instincts -- the ones that most resemble her "mad" relatives -- has long been lurking. In "A Knight of the Seven Kingdoms" she makes an effort to quell her most rash impulses, but how will she feel when the dust settles from the battle soon to come? Whatever she decides to do will likely dictate who comes out on top if and when the living take down the Army of the Dead.