Some writers might do it to create a sense of shock. They want to break the traditional view of the Dumbledore - Harry relationship and get readers' attention. It's a way to be different in the vast sea of fanfiction.
Well, it could be for the sake of creating a unique plot. Maybe the writer wants to show a different version of Harry's journey. By having Dumbledore as an abuser, they can introduce new conflicts and character developments. It might also be a way to comment on the idea that even the most respected characters can have hidden, negative aspects in a fictional and extreme way.
One possible motive could be to explore a dark and alternate universe. The writer might want to see how the 'Harry Potter' world would change if such a drastic event occurred. Another reason could be to analyze the characters from a different perspective. By making Harry the destroyer of Dumbledore, it forces readers to re - evaluate both characters.
It could be that Dumbledore is afraid of what Harry might do if left to his own devices. Harry has a lot of power, and Dumbledore may worry that he could be swayed by the wrong people or ideas. So, in an attempt to keep Harry on the 'right path', he resorts to trying to control him. But this also shows a lack of faith in Harry's character and judgment.
Maybe Dumbledore believes that Harry has some hidden power that is being corrupted by the environment at Hogwarts. So, he kidnaps Harry to a secluded place to train him and help him control this power without any distractions or negative influences from other students or even some of the teachers.
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.
In a fanfiction, Dumbledore might steal out of desperation. Suppose there was a situation where a curse was about to be unleashed on Hogwarts, and the only way to stop it was by stealing a particular ingredient for an antidote from a place where it was not supposed to be taken. Dumbledore, seeing no other option, decides to steal it. His motive here is to save the school and its students, even if it means breaking the rules.
In a 'Harry framed Dumbledore' fanfic, a motive could be revenge. Suppose Dumbledore made a decision that Harry thought was wrong and cost someone Harry loved dearly. Harry could then be driven by revenge to frame him. Also, it could be a case of Harry being under the influence of a dark force. The dark force might convince Harry that Dumbledore is a threat and needs to be removed from power, leading Harry to frame him.
One possible motive could be that Dumbledore saw a future where Harry's parents' deaths were necessary for a greater good. Maybe he knew that Harry would be the only one to defeat Voldemort in the end, and their deaths were a catalyst to set Harry on that path.
In some fanfictions, Snape could have his own personal vendetta against Dumbledore. Maybe Dumbledore has wronged Snape in the past, and when he sees Harry as a potential victim of Dumbledore's schemes, he decides to help Harry. Another reason could be that Snape has a new understanding of the prophecy and believes that Dumbledore's way of handling it is wrong, so he steps in to help Harry.
Perhaps in this fanfiction, Dumbledore has a secret plan related to Harry's power. He might think that by kidnapping Harry, he can train him in isolation to fully develop his magical abilities without the distractions and potential dangers at Hogwarts. It could also be that Dumbledore is trying to hide Harry from Voldemort in a way that no one would suspect, as Voldemort would never think Dumbledore would use such extreme measures against Harry.
One possible motive could be a misguided sense of protection. Dumbledore might think that taking something from Harry, like a powerful magical item, would keep it out of the wrong hands. He may believe that Harry is too young or inexperienced to handle it properly.