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.
In some fanfictions, it might be because Snape exposed a secret of Harry's that led to his departure.
One possible reason in fanfiction could be that Snape's constant bullying and unfair treatment finally pushed Harry over the edge. Harry might have felt that he couldn't take it anymore, especially if Snape made his life at Hogwarts unbearable in ways that the other teachers didn't notice or couldn't stop.
One possible reason could be that he feels he has fulfilled his purpose there. Maybe he has achieved certain goals related to protecting Harry Potter secretly, and he decides it's time to move on to other things. For example, in some fanfictions, he might leave to explore his own family history or to further his research on the Dark Arts in a more secluded place.
In fanfiction, one reason could be his desire for a fresh start away from the memories at Hogwarts. Maybe he feels he has done all he can there and wants to explore new magical knowledge elsewhere.
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.
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.
Well, Harry might become a much more serious student. Snape, being strict, could push Harry to excel in Potions and Defence Against the Dark Arts even more. He might also instill in Harry a sense of discipline that he didn't have before. And Harry could gain a different perspective on Snape, perhaps seeing him as a father figure over time.
If Snape kidnaps Harry in fanfiction, it could be because of a long - held grudge against James Potter. Since he can't get to James, he decides to take out his anger on Harry. But this would be a very dark turn for Snape's character.
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.
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.