Draco may save Harry from Dumbledore because he has come to understand Harry better. In the context of a romance fanfic, as they get closer, Draco realizes that Harry is not the enemy he once thought. Dumbledore's actions towards Harry might seem unjust to Draco, so he intervenes. Also, Draco could be motivated by a desire to prove himself to Harry. By saving Harry from Dumbledore, he shows that he is on Harry's side and worthy of Harry's love.
One motive could be love. In a romance fanfic, Draco's feelings for Harry might be so strong that he is willing to go against Dumbledore. He doesn't want to see Harry hurt by anyone, including Dumbledore. Another motive could be redemption. Draco might be trying to atone for his past wrongs and sees saving Harry as a way to do that.
There are many 'Draco Saving Harry' fanfics out there. Some explore the idea of Draco having a change of heart due to a shared danger they face, like a dark wizard threat that only they can overcome together. It's interesting to see how the author builds the trust between them in these stories.
I recommend 'A Different Kind of Savior' where Draco uses his knowledge of dark magic to save Harry from a death - eaters' trap. It's a short but intense read.
Perhaps Harry has been misled by someone he trusts. They could have filled his head with lies about Dumbledore, making Harry think that Dumbledore is actually a threat. And his motive would then be to protect the wizarding world from what he believes is a hidden danger in Dumbledore.
He could be doing it out of a sense of justice. Despite his family's views, Draco might see that the way the Dursleys treat Harry is wrong and decides to step in.
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 for Harry's protection. Dumbledore might have seen a threat that Harry wasn't aware of, and he asked Hermione to keep an eye on him. Hermione, being the loyal friend and also having great respect for Dumbledore, would have agreed to it.
In a romance fanfic scenario, Draco could save Harry from Dumbledore if there is some sort of misunderstanding. Dumbledore might be acting on information that Draco knows is false. Since Draco has fallen for Harry, he steps in to prevent harm. For example, Dumbledore could be about to send Harry on a dangerous mission that Draco knows is a trap. Draco's love for Harry makes him take a stand against Dumbledore.
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.
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.