Perhaps Jarvis hates it because it can create false expectations or rumors about the Avengers. Since he is part of their inner circle in a way, he doesn't want inaccurate ideas to spread. For instance, if a fanfiction implies a romantic relationship between two Avengers that is completely untrue in reality, it could cause unnecessary confusion.
In some fanfictions, Peter could have a secret that he's keeping from the Avengers. When it's eventually revealed, they might feel betrayed and turn against him. It could be something related to his powers, or his past associations that the Avengers find unacceptable, thus resulting in their hatred.
Differences in ideology can also play a part. Tony might have some views on how to handle threats or use technology that the other Avengers don't agree with. Maybe he's more willing to take risks with untested tech that could endanger the team or civilians, which would lead to resentment.
Perhaps Tails feels that fanfiction undermines the hard work of the original creators. He might think that the official story should be the only source of information about him and his adventures, and fanfiction is an unwanted deviation from that.
One possible reason is that Peter's quick rise to popularity among the public might make the Avengers jealous. They've been around for a long time, and suddenly this young kid is getting all the attention.
Jiraiya was a serious shinobi with a certain code. Fanfiction might play around with that code too much for his liking. It could make light of things he held in high regard, like his loyalty to the Leaf Village or his training of Naruto. So that's a possible reason for his hatred.
One reason could be that it doesn't respect the original story's intention. The original Naruto story is about growth, friendship, and ninja skills. Harem fanfiction for Team 7 just throws in romantic elements that Naruto may find disrespectful to the real story.
Perhaps Percy hates it because it creates a lot of plot holes and inconsistencies in the world he knows. If there's a son of Artemis in fanfiction, it might mess up the way the gods' relationships and powers work in the original story.
One reason could be a conflict of interests. If they are competing for the same thing, like a magical artifact or a position in a magical society, and Susan uses underhanded tactics, Harry would hate her.
One reason could be competition. If Chloe is presented as a rival in the fanfiction, Ash might hate her for trying to outdo him in Pokemon battles or other aspects. For instance, if she constantly shows off her skills or Pokemon, it can make Ash feel annoyed.
One reason could be that he was treated unfairly by other shinobi. If he was constantly looked down upon or discriminated against by his fellow shinobi, it might make him develop a hatred for them.