He'd probably be devastated at first. Gryffindor has been his home at Hogwarts, and his friends are there. But he's also a resilient character. He might start to question the fairness of the banishment and try to prove his innocence. He could look for allies outside of Gryffindor, like Luna from Ravenclaw, who might believe in him.
In some fanfictions, it could be a political move within Hogwarts. There might be a group that wants to undermine Dumbledore's influence, and since Harry is so closely associated with him, they target Harry by getting him banished from Gryffindor. This could also be related to the discovery of some long - hidden secret about Harry's family that makes the Gryffindors view him differently.
Another reason could be related to a new prophecy. Suppose a new prophecy emerged that said Harry Potter's presence in Gryffindor would bring great danger to the house. The Gryffindor students and faculty, believing in the importance of protecting the whole house, might have decided to banish him. But of course, this would be a very difficult decision as Harry has been an important part of Gryffindor before. And in the end, the prophecy might turn out to be misinterpreted.
In some fanfictions, Harry could become withdrawn and depressed at first. Losing his place in Gryffindor is like losing his family. But then, he might find new strength and purpose, perhaps by joining forces with some outcasts in the wizarding world and forming a new group that challenges the unjust actions of Gryffindor house.
In some fanfictions, it could be due to a misunderstanding. Maybe someone framed Harry for a serious rule violation that Gryffindor's house values highly, like cheating in an important magical exam. And the evidence against him seems so strong that the Gryffindor house decides to banish him.
He might become more independent. Without the support of Gryffindor, he has to rely solely on himself. So he becomes more self - reliant and his decision - making skills improve.
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.
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 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.