Another possibility is that he discovers some dark secrets about the wizarding world or Hogwarts itself. These secrets could be so disturbing that he feels he can't stay there any longer. It could be something related to the Ministry of Magic's true intentions or a hidden power within the school that he fears might be misused.
In fanfic, one reason could be that he might feel constantly in danger at Hogwarts. With all the dark forces targeting him, he could think running away is the only way to keep his friends safe. For example, if Voldemort's followers keep infiltrating the school, Harry might feel responsible and decide to leave to draw the danger away from others.
He could run away because he's tired of the expectations placed on him. Everyone always expects him to be the hero and defeat Voldemort. This pressure can be overwhelming. Maybe in the fanfic, he just wants to escape all that and find a normal life for himself, away from the wizarding world's chaos.
Maybe he wants to escape the constant danger in the UK wizarding world. Voldemort's followers might still be a threat even after his defeat, and France could seem like a safe haven.
One possible reason could be that he feels unsafe at Hogwarts. Maybe there are threats from dark wizards within the school that the teachers are unable to fully protect him from. Sirius, being his godfather, represents a safe haven.
Perhaps he runs away because he feels misunderstood. In the fanfiction, the other students or even some of the teachers might wrongly accuse him of something. He could be tired of constantly having to defend himself. Like if he was wrongly blamed for a serious incident at Hogwarts, and no one seemed to believe his innocence, he might just leave in frustration.
Well, it could be about them running away to avoid some dark magic threat at Hogwarts. Maybe a new dark wizard has emerged and they don't trust the school's ability to protect them at that moment.
Maybe they run away because they are tired of the constant danger at Hogwarts. They've faced so many threats, and they just want a break. So they run to a peaceful place where no one knows them.
Perhaps in the fanfiction, Harry Potter runs away from Britain because he is disillusioned with the way the Ministry of Magic operates. There could have been a major event where the Ministry fails him or betrays his trust. Also, he may be seeking a place where he can explore his magic without the constraints and expectations that come with being 'The Boy Who Lived' in Britain.
One reason could be that she has big dreams that can't be fulfilled where she is. Another might be that she's running from a person who's threatening her. Maybe she just wants to explore the world and find herself.
In a 'Harry and Hermione run away fanfic', they could be running away from the pressure of their pre - destined roles. Harry is always seen as the 'Chosen One' and Hermione as the brilliant sidekick. Fed up with these expectations, they decide to run away and explore the wizarding world on their own terms. They might travel to different wizarding schools across the globe, learning new spells and making new friends outside of Hogwarts.
One reason could be to escape the public spotlight. After the war, Harry is a hero and Ginny is his girlfriend. They might feel overwhelmed by all the attention and want a quiet life, so they run away.
One reason could be to escape the constant media attention and pressure that comes with being in the wizarding world's spotlight. Harry especially has always had the eyes of the public on him, and running away might be a way to get some peace.