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.
Perhaps Dumbledore was being controlled or influenced by some dark magic or entity that made him do it. He could have been in a trance - like state and not fully in control of his actions. Another motive could be that he was trying to protect a bigger secret, and Harry's parents were a threat to that secret. He might have thought that by killing them, he could keep this secret safe and ensure the future of the wizarding world in his own way, although this would be a very dark and morally ambiguous reason.
In a 'Dumbledore killed Harry's parents' fanfiction, a possible motive could be related to a power struggle. Dumbledore might have feared that Harry's parents, especially James, had too much influence or knowledge of a certain power or prophecy. He could have seen them as a potential threat to his own plans for the wizarding world. Maybe he thought that with their deaths, he could shape Harry into the tool he needed to bring about the kind of change or stability he desired, which is a very complex and controversial motive.
If such a fanfiction exists, perhaps the motive could be a misdirected attempt at protecting the greater good. Maybe Dumbledore thought Hermione's parents were in some way a threat to the wizarding world's secrets that he was trying so hard to protect. But this is still a very far - fetched idea as Dumbledore was always about protecting life in the official story.
Yes, there are likely fanfictions out there with this dark and alternative take. Fanfiction often explores non - canonical ideas, and this would be a very radical one. It goes against the established lore in the official 'Harry Potter' series, but that's the nature of fanfiction - to play with different concepts.
Well, it could be for the sake of exploring a different kind of magic or a different set of rules in the wizarding world. Perhaps in this fanfiction's version, the parents were under some sort of dark magic influence that made them turn on Harry. Or it could be a psychological exploration. The writer might be interested in delving into the minds of Harry's parents and seeing what could make them do such a heinous act. It could also be just a very different take on the concept of family and love in the 'Harry Potter' universe.
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.
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.
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.
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.