Sure. In some fanfictions, Dumbledore might teach Harry about leadership. Given that Harry has a big role to play in the wizarding world, Dumbledore could impart wisdom on how to lead others, make difficult decisions, and inspire loyalty. This kind of teaching can be crucial for Harry's development as a character in the fanfiction world.
It has a huge impact. First, in the context of the fanfiction, Dumbledore's teaching of Occlumency could be a way for Harry to finally have some privacy in his mind. Harry has always been an open book in some ways, and this would change that. Second, it could enhance his relationships. He'd be less likely to accidentally project his thoughts or feelings onto others. And third, it would give him an edge in the fight against the Dark Arts. He could better resist Legilimency - based attacks and perhaps even use Occlumency in a more offensive way against the Death Eaters.
He could gain the ability to shield his mind completely. This means no one, not even the most powerful Legilimens like Voldemort, could peek into his thoughts. Also, he might be able to project false thoughts, a useful skill when dealing with enemies. For instance, he could make them think he's planning one thing when in fact he's doing another.
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.