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.
It could be for love. They might want some time alone, just the two of them, without any distractions. In fanfiction, the author could be exploring a more intense and private relationship between Harry and Ginny. They may feel that the only way to truly be together and grow their relationship is to run away from all the responsibilities and expectations that are placed on them in the wizarding society. For example, they could be tired of dealing with the politics and social norms that might be getting in the way of their relationship, so they choose to run away and start anew.
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 possible plot could be that they run away to escape some dark magic threat that Voldemort's remaining followers are planning. They might go into hiding in a remote wizarding village, where they have to learn to fend for themselves without the help of the Order of the Phoenix. Along the way, they could discover new magical abilities or ancient spells that help them in their fight against the dark forces.
Maybe they are in love and their families oppose their relationship. So, they run away to be together.
Maybe Harry feels a sense of guilt. In the context of the war, he might think that he's failed in some way by getting into this unexpected situation. He could believe that his new condition might put others at risk or that he's no longer the hero everyone expects him to be. So, to protect others or to avoid the shame, he decides to run away.
Well, one possible plot could be that they run away to escape some sort of dark threat at Hogwarts. Maybe a new dark wizard has emerged and they feel that the school is no longer safe. So, they secretly gather some supplies, use magic to disguise their departure, and head off to a remote location like a hidden wizarding village in the mountains. There they try to find a way to fight the threat from afar while also exploring their relationship in a more private setting.
One possible reason could be a growing apart due to their different career paths. Harry, being deeply involved in the wizarding world's security and Ginny having her own Quidditch career. Another reason might be the stress of living in the public eye. Their relationship may not be able to withstand the constant scrutiny and pressure from the wizarding community.
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.
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.
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.