At first, it would become very strained. Harry might avoid Lupin at all costs. He would no longer trust Lupin and would be suspicious of his every move.
One possible event could be that Lupin lied to Harry about something important. In the world of Harry Potter, trust is a huge factor. If Lupin deceived Harry, it could easily turn Harry against him. Say, Lupin lied about his whereabouts during a crucial time when something bad happened, and Harry suspected Lupin's involvement.
One possible reason could be a misunderstanding. Maybe Lupin did something that Harry misinterpreted as betrayal or negligence. For example, if Lupin was forced to keep a secret from Harry for his own safety, but Harry found out in a way that made him feel deceived.
One great 'Harry Potter - Arsene Lupin' fanfic is 'Lupin's Legacy'. It delves deep into Lupin's past and his relationship with the other characters. It shows how his werewolf nature affected his life at Hogwarts and beyond in a really interesting way.
Maybe in that fanfic, Lupin did something that Harry considered a betrayal. For example, he might have made a decision that put Harry or his friends in danger without consulting them first.
Character development makes it interesting. In a good fanfic, Lupin's complex character is explored further. His struggle with his werewolf identity, his loyalty to his friends, and his role as a teacher all can be delved into more deeply than in the original books.
There may be some fanfics out there with this premise, but it goes against the general relationship dynamics in the official 'Harry Potter' universe. Usually, Harry respects and values Hermione. If such a fanfic exists, it might be written by someone who wants to explore a really dark or different side of the story, like if there was a major event that made Harry turn against Hermione in that fictional narrative. But I haven't actually come across one myself.
One possible reason could be a miscommunication. Maybe Hermione accidentally reveals a secret of Harry's that he didn't want known, and this causes him to feel betrayed and start to hate her. Another reason could be jealousy. If Hermione gets more attention or praise for her magical abilities than Harry in the fanfic, he might develop feelings of envy and hatred. Or perhaps there's a love triangle situation where Hermione is involved with someone Harry doesn't like, and that causes his hatred.
One possible reason could be a misinterpretation of his character. In the original series, Harry Potter doesn't hate Muggles at all. But in this fanfic, perhaps the author wants to explore a 'what if' scenario where something drastic has happened to make him have such feelings. Maybe he's been betrayed by a Muggle - in the fanfic world, this could be a made - up situation that sours his view on all Muggles.
This type of fanfic might shift the focus from the fight against Voldemort to the internal conflict between Harry and Dumbledore. It could explore different character traits of both of them that were not as prominent in the original. For example, Harry's potential for anger and Dumbledore's more flawed side.