He might start to question some of the norms in his own wizarding world. If he reads about how the characters in Supernatural deal with curses and dark forces without the use of spells like in his world, he could start to wonder if there are better ways in his own society. It could make him more critical of some of the traditional ways in Hogwarts and the wizarding community at large.
His view could become more open - minded. He'd realize there are other ways to approach the supernatural world. For example, the way the hunters in Supernatural use weapons instead of wands might make him think about alternative forms of magic defense.
His view of magic could expand. Anime often has very different interpretations of magic and superpowers. He might start to think more broadly about what magic can do and how it can be used.
Their view might expand. They'd see different takes on the wizarding world that they hadn't considered before.
His view of magic might expand. In his world, magic is mainly about spells and potions. But after reading HTTYD fanfiction, he would see that magic can also be about the bond between humans and creatures like dragons. This could make him think more about the potential of magic in his own world beyond just what he has been taught at Hogwarts.
His view could become more broad. He'd see that magic can be manifested in ways other than what he's used to. For example, the zanpakuto in Bleach have unique powers that are different from any wand magic he knows.
Their view could become more confused. Fanfiction can present all sorts of alternate universes and character developments. The Death Eaters, used to a very specific view of the wizarding world under Voldemort, would find it hard to accept different versions. They might see the wizarding world in fanfiction as a chaotic and wrong place, which would make them double down on their own views to feel more secure.
After reading the fanfiction, Percy could have a more in - depth understanding of the wizarding world. He might start to think about how the social structure in the wizarding world compares to that in his demigod world. He might also be more curious about the history and lore of the wizarding world, and how it developed compared to his own world of Greek gods and monsters. He could even start to imagine what it would be like to be a wizard, with all the unique spells and magical items, and how that would change his approach to dealing with his own problems.
Dumbledore could gain new insights. Fanfiction might explore the social aspects of the wizarding world in ways he hadn't considered. Maybe there are stories about how the different houses interact in a more complex way than he thought. This could make him reevaluate his own views on the unity or division within Hogwarts.
If Harry invented spells like 'Teleporto Instanto', which allows for instant teleportation anywhere in the wizarding world, it would completely revolutionize travel. People could go from Hogwarts to Diagon Alley in a blink. It would also change the way battles are fought as wizards could quickly escape or surprise their enemies.
They may gain a deeper understanding of how the outside world perceives their world. Fanfiction often reflects the hopes, dreams, and fears that fans have about the wizarding world. So, by reading it, the Potter family can get a sense of what others find interesting or concerning about their world. For instance, if a lot of fanfiction focuses on the social inequalities in the wizarding world, it might make them more aware of these issues and perhaps even inspire them to do something about it.
The wizarding world would likely be plunged into a state of despair. Harry was seen as the 'Chosen One' to defeat Voldemort, so without him, there would be a sense of hopelessness. Voldemort would probably have free rein to impose his dark rule without much opposition at first.