Often in these fanfics, Sakura might make fun of Naruto's dreams or goals in a very mean - spirited way. This constant put - down could lead Naruto to hate her. Also, if Sakura were to side with Naruto's enemies in the fanfic for some reason, like being brainwashed or having a hidden agenda, it would be a likely cause for Naruto's hatred.
One possibility for such a fanfic is that Naruto has been through some kind of trauma that makes him view Sasuke and Sakura differently. For example, if Naruto was betrayed by someone he trusted deeply who had similarities to Sasuke and Sakura in some aspects. It could change his perception of them completely and fuel his hatred in the fanfic.
In fanfiction, the evidence often lies in the way the author portrays Naruto's interactions. For example, Naruto may be shown being cold or dismissive towards Sakura, like avoiding her company or not showing any romantic interest. Meanwhile, with Hinata, he could be depicted as being more affectionate, like blushing around her or always coming to her aid. But it's important to note that this is all in the realm of fan - created interpretations and not the official canon.
In such a fanfic, one possible reason could be Naruto's feeling of being left out. Maybe he sees Sakura's constant focus on Sasuke and feels neglected in their friendship triangle. Another reason might be jealousy. If Naruto has unrequited feelings for Sakura, seeing her love for Sasuke could turn to hatred over time.
The relationship would become very tense. Naruto would likely avoid Sakura and Sasuke. There would be no more of the friendly banter or the complex emotional triangle that was there before. It's a complete shift from their normal relationships in the original story.
Yes, there probably is. In the Naruto fandom, there are various fanfiction themes, and the 'narusaku' (Naruto - Sakura relationship) with Naruto hating Sakura can be an exploration of an alternative take on their relationship. Some fans might write this to explore what - if scenarios or to express their own views on the characters' dynamics that deviate from the canon.
One possible reason could be for the sake of creating a more complex love triangle in fanfiction. Maybe the writer wants to explore a different dynamic where Naruto and Hinata's relationship is strengthened by having a common dislike for Sakura. It could also be that the fanfiction writer is trying to give Hinata more agency and make her a stronger character by having her oppose Sakura in some way, perhaps in matters related to Naruto's affections.
It could also be a misunderstanding. Perhaps Karin is misjudged by Sakura in the fanfic. Maybe Karin did something that was misinterpreted as malicious towards Sakura or someone she cares about, leading to the hatred.
Maybe in the fanfic, Sasuke has done something to betray Sakura's trust. For example, he might have left the village again without considering her feelings.