There could be many reasons. Maybe some fanfiction writers want to create an alternate universe where Harry has a different path. It could also be for the sake of creating a more complex and unexpected plot, like having him be falsely accused of something so bad that Gryffindor doesn't want him. Another possibility is that the writer wants to explore how Harry would react and grow outside of the Gryffindor environment he's so closely associated with in the original series.
He might feel a great sense of betrayal. Since Gryffindor was like his home in Hogwarts, being banned could make him question his identity. He could become more isolated and withdrawn at first. However, this could also lead to him finding new friends and allies from other houses or even outside of Hogwarts.
Yes, in some fanfictions, Harry Potter is depicted as the heir of Gryffindor. It adds an interesting twist to the original story. The connection to the Gryffindor sword becomes more profound as it is often a symbol of Gryffindor's legacy. In these fanfictions, Harry might discover hidden powers or a deeper understanding of his destiny through this new identity.
One possible reason could be copyright issues. The original creators and publishers might want to have strict control over the Harry Potter universe to protect their intellectual property. They may be concerned that fanfiction could potentially distort or misuse the original concepts and characters in ways that are not in line with their vision for the franchise.
One possible reason could be a false accusation of using dark magic. In some fanfictions, the wizarding world can be very paranoid about dark magic, and if Harry was wrongly suspected of delving into it, Gryffindor might exile him to protect their reputation.
One possible plot could be that Harry is framed for something he didn't do. Maybe a powerful dark object is found in his dorm, and the Gryffindor house, fearing for their reputation and safety, decides to exile him. Harry then has to prove his innocence while also dealing with the isolation and new challenges outside of his familiar house.
Often, it could be a plot device. For example, it might be the key to unlocking an ancient mystery in the RWBY world. Since the Sword of Gryffindor only appears to those worthy in the Harry Potter series, in RWBY fanfic, it could be that only a truly noble character can access it. And when they do, it causes a stir in the RWBY world as it's an unknown and extremely powerful object.
There could be several reasons. One is simply to add drama. A lot of fanfiction writers thrive on creating conflict, and what better way to do that than by taking on one of the most beloved houses in the 'Harry Potter' series? Another reason might be personal bias. Perhaps the writer had a negative experience with someone who identified strongly with Gryffindor, or they just don't like the traits that are typically associated with Gryffindor, like impulsiveness, and they use fanfiction as a way to express their views.
One great 'Harry Potter Gryffindor Fanfiction' is 'The Gryffindor Princess'. It has a really interesting plot where a new student arrives at Hogwarts and is sorted into Gryffindor, but she has a mysterious past that ties into the original story in unexpected ways.
There could be a fanfiction where a Gryffindor Animagus is trying to find out more about their ability. They might search through the restricted section of the library. In their Animagus form, they could access areas that are hard to reach for normal students. And they might meet other magical creatures who can guide them in understanding their powers better.