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.
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.
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.
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.
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 reason could be to search for a powerful magical artifact outside of Hogwarts that they believe is crucial for defeating a new dark threat. In fanfiction, they might feel that Hogwarts has become too dangerous or restricted in terms of resources for their mission.
Maybe he leaves because of a conflict with the Hogwarts staff. If the new headmaster or some teachers start implementing rules that Harry believes are wrong or unfair, like restrictions on defensive magic classes, he might decide to leave in protest.
It could also be due to personal reasons related to his growth and self - discovery. The events of the Triwizard Tournament might make him realize that he needs to find his own path outside of Hogwarts for a while. He might feel that the school can't offer him the answers or the experiences he needs at that moment to become stronger and more independent.
It could also be that Snape represents everything that Harry hates about Hogwarts at that moment in the fanfiction. Snape's strictness, his partiality towards certain students, and his general unfriendly attitude might make Harry feel that Hogwarts is no longer the safe haven it once was. So, he leaves, thinking that he can find a better place where he doesn't have to deal with Snape's negative presence.