He might do it because the Daily Prophet has been printing untrue stories about him.
There are several possible reasons. First, the Daily Prophet might be under the control of the dark forces or Voldemort's supporters in the fanfic. Harry, being the hero, wants to cut off this source of misinformation and propaganda. Second, they could have printed some secrets that are crucial for the safety of the wizarding world, like the location of a safe house or the plans of the resistance. Third, the paper may have slandered his friends and family, and Harry, with his strong sense of justice, cannot let this go unpunished. So, he takes the drastic step of destroying the Daily Prophet.
Well, perhaps Harry is fed up with the false or malicious stories in the Daily Prophet in the fanfic and decides to take action to destroy it. Maybe he uses magic to burn all the copies or disrupt the printing process.
Hermione would most likely support Harry. She's always been his loyal friend and would see the injustice of the Daily Prophet's actions. She could help him research laws in the wizarding world (if there are any regarding slander or privacy) and also use her intelligence to find ways to counter the newspaper's claims.
Maybe Harry sues the Daily Prophet for spreading false rumors about him, like making up stories that he's secretly working for Voldemort again. It could be that these false reports are ruining his reputation and causing problems in his relationships at Hogwarts and in the wizarding community.
They might have to issue a public apology. This is a common consequence when sued for something like false reporting. It would be a way to try and repair some of the damage done to Harry's reputation.
He might be interested in the articles about new magical creatures. Since he has had many encounters with different and often dangerous creatures, news about newly discovered ones or updates on known ones would surely catch his attention.
Maybe he wants to keep up with the latest news in the wizarding world. There could be important announcements, like news about Voldemort's activities or updates on the Ministry of Magic.
If Percy is a character, he might destroy the fanfic out of jealousy. Maybe the fanfic is getting more attention than he is used to, so he takes actions to undermine it.
In a fanfic scenario, the photo in the Daily Prophet might be misconstrued. Suppose it's a photo where Hermione and Ron are just having a friendly chat, but the Prophet spins it as a romantic moment. This could lead to all sorts of drama. Ron might be embarrassed and Hermione could be angry at the false portrayal. Their friends would then have to step in to clear things up.
Well, it might be that the Goblet of Fire has been corrupted in the fanfiction. It could be causing chaos in the Triwizard Tournament, endangering the lives of the students more than it should. Harry, being the hero, takes it upon himself to destroy it to restore order. For example, if it starts spawning dangerous and unfair tasks that are clearly not part of the original tournament's design.
Ash might destroy Tobias fanfiction out of envy. If the fanfiction is really popular and Ash wants that attention for themselves.