It might also mean that the fight against Voldemort takes a different turn. If Harry leaves Hogwarts, he may not be as well - equipped with the magic knowledge and skills he would have gained there. So the battles against the Death Eaters could be much more difficult and have different outcomes.
Well, it could completely disrupt the relationships. For instance, his friendship with Ron and Hermione might be strained as they try to convince him to stay. This is different from the original where they are all united in their Hogwarts experience.
One possible reason could be that he feels constantly in danger at Hogwarts. With all the dark magic threats and attacks that keep happening, he might think leaving would keep his friends and himself safer. For example, after the Chamber of Secrets incident, he might have considered leaving to protect others from Voldemort's influence.
The fight against Voldemort would be very different. Since Hogwarts is a key place in the battle against the Dark Lord, if Harry isn't there, the defenses of Hogwarts might not be set up in the same way. Also, he won't be trained by the professors, so his magical skills won't develop as they did in the original story. Without Harry at Hogwarts, the balance of power between good and evil in the wizarding world could shift in a very unexpected way.
He might leave Hogwarts in fanfic due to a sense of disillusionment. Maybe he discovers some dark secrets about the school or the wizarding world in general that make him question his place there. It could be something related to the way the school is run or a conspiracy among the teachers that he stumbles upon.
One possible reason could be that he's tired of the constant danger at Hogwarts. There are always threats like Voldemort's followers or dark creatures lurking around. Another reason might be that he feels pressured by the expectations placed on him as 'the Chosen One'. Everyone expects him to save the day all the time, and it's a huge burden.
His decision to leave the Dursleys can make the plot deviate in many ways. In fanfic, it could mean that he doesn't have to wait for the official Hogwarts invitation. He might seek out the wizarding world on his own. This could lead to him encountering different characters or facing different challenges earlier. Also, his departure might make the Dursleys more or less relevant in the overall story. If they are no longer a factor, then new antagonists or plot elements need to be introduced to keep the story interesting.
A fanfic with Cedric and Harry as brothers would change the Hogwarts experience for both of them. They could share a dormitory or have special family - related privileges at school. Their bond as brothers would likely make them more of a united front against the likes of Voldemort. Instead of Harry being the sole focus of Voldemort's attention, Cedric would also be in the crosshairs. This would lead to different strategies and plot developments. The Weasley family might interact with them differently too, as they would be related to Harry in a different way. And the relationship between Harry and Cho Chang (who was Cedric's girlfriend) would be completely reimagined, adding a whole new layer of complexity to the story.
The story could take a darker turn if he refuses to go back to the Dursleys. Maybe without the protection of the blood magic at the Dursleys' place (even if they were cruel), he becomes more vulnerable to Voldemort's attacks earlier. Or, on the other hand, he could find a new form of protection that was not part of the original story, which could change the power dynamics in the fight against Voldemort.
The teachers would also be concerned. Dumbledore might try to have a deep conversation with Harry to understand his motives. Some teachers might see it as a sign of weakness or that they have failed in some way to make Hogwarts a comfortable place for Harry. As for his enemies, they might see it as an opportunity to strike while he's vulnerable.
This type of fanfic might shift the focus from the fight against Voldemort to the internal conflict between Harry and Dumbledore. It could explore different character traits of both of them that were not as prominent in the original. For example, Harry's potential for anger and Dumbledore's more flawed side.
Well, it changes everything. Without Harry, there's no 'Chosen One' to fight Voldemort. So, Voldemort would likely rule the wizarding world unopposed. The Hogwarts School might become a place that only teaches dark magic or is completely under Voldemort's control.