In Harry Potter fanfiction, a male professor can play various roles. For example, he could be a mentor figure like Dumbledore, guiding the young wizards. He might also be a villainous professor, creating obstacles for the main characters. Some fanfictions might explore a male professor's backstory, his relationship with other characters in the wizarding world, or his own magical research and discoveries.
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.
Well, in this fanfiction, Harry might be a professor because of his extensive knowledge and experience in magic. He could be teaching younger students and sharing his wisdom.
Adventure is also a popular theme. They might go on a quest together, perhaps to find a lost magical artifact or to solve a mystery that has been plaguing Hogwarts for years. Their combined skills - Hermione's book - smarts and Lupin's practical experience - make them a great team for these adventures.
You can start by looking on fanfiction websites like Archive of Our Own (AO3). There are a large number of Harry Potter fanfictions there, and you're likely to find some that involve Hermione and Professor Lupin. Another option is Fanfiction.net. Just use their search function and type in relevant keywords like 'Hermione Lupin' to narrow down the results.
One interesting fanfiction idea could be that Snape discovers a hidden Horcrux that Lupin unknowingly has in his possession. Harry Potter then has to figure out how to destroy it without harming Lupin. This would add a whole new layer of drama and mystery to the existing lore of the Harry Potter world.
One great fem Harry Potter and Remus Lupin fanfiction is 'A Different Path'. It explores an alternate universe where Harry is female and her relationship with Remus develops in a very interesting way. It delves deep into their characters and how they interact in a world full of magic and mystery.
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.