There are several potential reasons. It could be due to the interference of a powerful magical being who wants to see how Harry will fare in the dangerous tournament. Maybe the Goblet of Fire has its own consciousness in the fanfiction and senses that Harry is a key figure in the wizarding world at that time. It might also be a result of some sort of magical glitch or misfire that, instead of following the normal rules, selects Harry because of his unique magical signature.
One possible reason could be the dangerous tasks in the Triwizard Tournament. The tasks are extremely challenging and life - threatening. For example, facing a dragon in the first task would be terrifying for anyone, let alone Harry who is just a student. Another reason might be the mystery surrounding the Goblet of Fire itself. It seems to have a mind of its own, choosing Harry when he didn't even enter his name, which could make him feel very uneasy as he is thrust into a situation he didn't expect.
Maybe it's related to his relationship with Cedric. In the Goblet of Fire, Cedric is a competitor but also a friend in a way. His death is a huge shock in the original story. In fanfiction, it could be that events leading up to Cedric's death or just the general unfairness of the situation makes Harry cry. For example, if there are false accusations against Cedric or if Harry feels he could have done more to prevent the tragedy.
One possible scenario could be that Harry is tricked by a dark wizard into believing that his friends are in danger if he doesn't betray the rules of the Goblet of Fire. Maybe he's shown false visions of Ron and Hermione being tortured, and in his panic to save them, he does something that goes against the fair play of the tournament, like getting inside information about the tasks through improper means. Another could be that he's under the influence of a powerful potion or spell that makes him act against his normal values and betray the trust placed in him regarding the tournament.
Another storyline could be that Harry is alone as he tries to break a curse that has been placed on him during the Goblet of Fire events. The curse is making him see things that aren't there and is slowly draining his magic. Since he doesn't know who to trust, he tries to find a cure by himself. He travels to different parts of Hogwarts and even outside the school grounds in search of answers, all while feeling the loneliness of being on his own.
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.
One possible scenario in such a fanfiction could be that Harry, perhaps under the influence of the dark magic or some sort of spell, starts to bully his fellow students during the Goblet of Fire events. Maybe he taunts Cedric Diggory out of jealousy for being chosen as a Triwizard champion. However, this would be a huge deviation from his normal character.
One interesting plot twist could be that Harry has an unexpected ally in the Triwizard Tournament. Maybe a Slytherin student who has a hidden agenda but ends up helping Harry in a crucial task. This would change the dynamic of the traditional House rivalries.
In some fanfictions, a great plotline could be Harry and Luna forming a closer bond because of the Goblet of Fire chaos. Luna could be the one to comfort Harry when he's facing the difficulties and dangers of the tasks. And together, they might even uncover a conspiracy among the teachers or judges that was meant to manipulate the results of the Triwizard Tournament.
One possible interesting story could be Harry's internal struggle that leads to him snapping at the Goblet of Fire. Maybe he's frustrated with the way the Triwizard Tournament is being handled or he suspects some foul play related to the Goblet's selection. It could also show a different side of Harry's character, like his growing impatience with the magical world's bureaucracy.
One interesting plot idea could be exploring what would happen if a different student was chosen for the Triwizard Tournament. Maybe a shy and overlooked student gets picked, and they have to overcome not only the tournament tasks but also their own insecurities. Another idea could be a love triangle involving characters that were not part of one in the original story. For example, a friendship between a Hufflepuff and a Slytherin turns into a complex relationship when feelings for Harry Potter get involved.