One possible reason could be his unruly behavior. Naruto is known to be a bit of a troublemaker in the series, always pulling pranks. In fanfiction, this might be exaggerated, and the orphanage might not be able to handle it anymore. Another reason could be that he's suspected of being involved in something dangerous or illegal, even if he's innocent. For example, if there are strange incidents around the orphanage and Naruto is the easy target to blame because of his reputation.
One possible reason could be his prankster nature. Maybe he played one too many harmless but annoying pranks on the staff or other orphans, and they decided they'd had enough. Another reason might be the fear of the Nine - Tails within him. Even if the people in the orphanage didn't fully understand what it was, they could sense something dangerous about Naruto and wanted to distance themselves from him for safety reasons.
Naruto might start to develop his ninja skills earlier. Without the orphanage, he has more freedom to explore and train. He could find hidden training grounds or meet mentors in unexpected places. His determination to prove himself and show that being kicked out was a mistake could fuel his growth as a ninja.
Naruto's life becomes a lot more difficult. He has to find shelter on his own. He might start sleeping in abandoned buildings or under bridges. He has to scrounge for food, which could lead him to steal at times. His loneliness intensifies as he no longer has even the meager company of the orphanage. He also becomes more distrustful of people, thinking that if the orphanage could reject him, so could anyone else.
Maybe he was blamed for some trouble he didn't actually cause. For example, if something went missing in the orphanage and the caretakers wrongly thought Naruto took it.
Naruto's view of the world and people around him might change drastically. He was already an outcast, but being kicked out of the orphanage could make him more distrustful. He'll have to learn to fend for himself, which could lead to him developing new skills. For example, he might become better at stealing food to survive, but at the same time, he could also meet some kind - hearted people who will help him and teach him better ways to live.
Naruto might initially be shocked and hurt. After all, being expelled by his own family is a huge blow. But being 'godlike', he could quickly turn that pain into determination. He might decide to prove his worth and show his family that they made a big mistake.
Well, one possible plot could be Naruto leaving Konoha and finding a new group of allies who are also outcasts. They then form a powerful team to prove themselves to Konoha.
Iruka Sensei is often one of the characters. He has always been like a father figure to Naruto to some extent. In these fanfictions, he may offer Naruto a place to stay and continue to train him. Another one could be Jiraiya. He might take Naruto under his wing more so than in the original story, seeing Naruto's potential and his difficult situation.