Yes, there are some fanfictions with such themes. However, these often deviate from the original relationship in the 'Harry Potter' series and can be quite controversial as it goes against the more mentor - mentee dynamic that was presented in the books.
In such fanfictions, a common aspect is the exploration of forbidden love. Since their relationship in the original is more of a teacher - student one, the fanfiction might play with the idea of breaking those boundaries. There could be a lot of internal conflict for Dumbledore as he grapples with his feelings. Also, there might be a change in how other characters perceive their relationship, which adds to the drama and the alternate universe feel of the fanfiction.
Dumbledore, as Harry's guardian in the fanfiction, would probably use his influence in the wizarding world. He could make sure that Harry is taught by the best teachers at Hogwarts. He'd also be there to offer advice and guidance, especially when Harry faces challenges like dealing with Voldemort or other dark forces. For example, he might teach Harry some advanced defensive magic privately to better protect himself.
Well, in some fanfictions, Harry might attack Dumbledore if he discovers that Dumbledore has been using him as a pawn in a bigger game all along. This feeling of betrayal could drive him to lash out. Also, if there is a prophecy that he misinterprets to mean Dumbledore is a threat to his future or the future of his friends, that could be a motive.
Snape can be a prominent character. Given his complex relationship with Dumbledore and Harry, his perspective could add a lot to the story. He might have some insights into Dumbledore's actions that Harry doesn't know about, and his reactions to Harry's blaming could vary from being scornful to having a moment of understanding. Luna Lovegood could also be a part of it. Her unique view of the world might make her offer some different viewpoints on the situation between Harry and Dumbledore.
Dumbledore might give Harry advanced spellcasting training. He could teach Harry some of the more complex and ancient spells that are not commonly taught at Hogwarts.
Hermione could be a main character. She might use her vast knowledge to help Harry and Dumbledore in their adventures. For example, she could research ancient spells or potions that are crucial for their mission.
Perhaps there are safety concerns for the students that Dumbledore is not addressing adequately. Harry, having a strong sense of responsibility for his fellow students, may feel that the only way to ensure their safety is to remove Dumbledore from his position. He might believe that Dumbledore's actions or inactions are putting the students at risk, and thus he takes matters into his own hands.
The relationship would become very strained. Harry's rejection would likely make Dumbledore re - evaluate his actions and how he has interacted with Harry. Dumbledore might be shocked at first, since he probably thought Harry was on the same page as him most of the time.
Well, it could be that Harry has a different view on how to handle the war against Voldemort. Dumbledore is often very strategic and sometimes keeps things from Harry for what he believes is the greater good. But Harry might think that full transparency is necessary. Also, Harry may have formed new alliances or friendships that give him different perspectives, and these new ideas conflict with Dumbledore's. He might feel that Dumbledore's approach is too old - fashioned or risky in a new context.