It could be in a fanfic where Harry teams up with someone powerful or uses a clever strategy that no one expected. The story could also change the nature of the task or the dragon itself to make it possible for Harry to succeed early on.
It could be an entirely new type of dragon created specifically for this fanfiction. The author might have wanted to introduce a unique dragon with special abilities, like a dragon that can change its color to blend in with its surroundings or one that can breathe not just fire but also some kind of magical ice. Harry would then have an even greater challenge when trying to kill it with the sword.
If Harry were a Dragon Animagus in the First Task, it would change the whole dynamic. He could use his dragon form to create a diversion. For example, he could make the real dragon chase him away from the egg, and then quickly transform back and snatch the egg. Also, his dragon form might have some special abilities like breathing fire in a controlled way to fend off the real dragon if it attacks.
One possibility could be that Harry, as a Dragon Animagus, uses his new form to his advantage in the First Task. He might be able to outmaneuver the real dragon more easily, flying around it and confusing it. This would give him an edge in retrieving the golden egg.
The playing of memories could also have a negative impact. If the memories are too distracting or if they make him overthink, it could lead to mistakes. For instance, if he gets too caught up in the memories of a champion who failed, it might make him nervous and cause him to make errors during the first task.
In certain 'Harry kills the dragon' fanfictions, Harry could be given a special magical weapon specifically designed to slay dragons. He might have to go through a series of trials to figure out how to use this weapon effectively. It could be that the weapon has to be powered by his own magic and emotions. Once he masters this, he is able to take on the dragon and kill it. For example, the weapon might need to be charged with his feelings of courage and determination, which are strong in Harry, and then he uses it to strike a fatal blow to the dragon.
It could be in a dark alternate universe fanfic where Harry is driven to extreme actions due to unforeseen circumstances.
Well, it would have a huge impact on the wizarding world as a whole. The Ministry of Magic would be in an uproar. They had high hopes for Harry and his role in the future of the wizarding community. His death could also lead to a change in the way Hogwarts approaches the Triwizard Tournament in the future. There would be a lot more safety regulations and restrictions. And his death would leave a big emotional void in the hearts of those who knew him, like Hagrid who had a special bond with Harry.
One possible consequence could be chaos at Hogwarts. The school has high hopes for Harry in the Triwizard Tournament. His death would shock the students and teachers. The other champions might feel guilty or be blamed. Also, Voldemort's plans related to Harry would be disrupted, as he was supposed to be a major part in the wizarding world's future battles against the Dark Lord.
One interesting plot could be a character having to train a dragon for the task in a completely new way, using magic that was previously undiscovered. This would add an element of mystery and innovation to the story.
In many fanfictions, Harry might use his powerful spells like the Stunning Spell or the Killing Curse if the story allows for such dark magic use. He could also use his intelligence to outwit the dragon, perhaps finding a weakness in its armor or behavior pattern and then using a well - timed spell to defeat it.