Well, it might show Harry on a journey of self - reliance. He could be traveling alone, facing new challenges outside of the school's protective bubble. Maybe he meets new allies who help him understand the bigger picture of the magical world that Hogwarts had shielded him from.
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 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.
Maybe the fanfiction portrays some really dark and disturbing events that make Harry feel he can't stay at Hogwarts anymore. For example, if it involves his friends being seriously harmed in the fictional story in a way that he can't bear to be associated with the place where it all seems to be happening.
One possible plot could be that Harry leaves Hogwarts to hunt down the remaining Death Eaters on his own. He might feel that he can't put his friends in danger anymore and takes it upon himself to rid the wizarding world of the last remnants of Voldemort's followers.
Neville Longbottom could be a key character. He might take on a bigger role in Hogwarts in Harry's absence and also be in touch with Harry, providing support from the school. Ginny Weasley would also be involved as she has a close relationship with Harry. Additionally, some of the Hogwarts professors like Professor McGonagall could play important parts, either trying to bring Harry back or helping him in his mission outside the school.
It could be Dumbledore in a lot of fanfictions. Dumbledore often plays a key role in these stories and his reasons for summoning Harry could vary. Maybe he wants Harry to prove himself in a different way, or he has some premonition about the tournament that only Harry can help with.
Hermione might go with him. She's always been by Harry's side, and she would likely follow him out of loyalty and also because she might see the value in whatever mission or journey Harry is embarking on. She could provide the intellectual support and research skills that Harry needs.
The students would surely miss her. She's a role model for many, especially the younger students who look up to her intelligence and determination. Without her, the study groups and academic competitions might not be as intense or successful.
It could be a new dark wizard or witch who has infiltrated Hogwarts in the fanfiction. This character might be causing all sorts of trouble like spreading dark magic among the students or causing unrest between different houses.