Well, if Naruto has a bloodline and hates Konoha in a fanfic, it could lead to him becoming an anti - hero. He might use his bloodline powers against Konoha at first. Maybe he teams up with some of Konoha's enemies who share his hatred for the village. This would completely turn the original story on its head.
One idea could be that as Naruto leaves Konoha, he encounters a powerful and ancient creature in the wild. In a battle with it, the extreme stress and need to survive unlocks his hidden bloodline. This new power could be related to some kind of elemental control that is unique to his clan, like a special form of wind manipulation that is far more powerful than what he had before.
The first consequence could be a breakdown in the village's unity. Naruto is seen as a hero in Konoha, and if he turns against it, many of his supporters might be confused or even follow his lead. This could lead to internal strife within the village.
One possible reason could be that Konoha has a history of dark secrets. Maybe Naruto and Kushina discovered some unethical practices within the village that made them lose faith in it. For example, the way the village treated the Uchiha clan was quite unjust. Kushina might have known about some of these underhanded dealings and passed on her concerns to Naruto.
Another reason could be that he witnesses corruption or unfairness within the village leadership. If he sees that the system is rigged against certain groups, including himself or his friends, he could grow to hate Konoha and decide to leave.
It could also be due to a lack of support. Naruto was constantly in trouble and trying to prove himself. If his family didn't show up in any way to support him and Konoha didn't do enough to encourage him in a positive way, but rather just punished him for his pranks, he might grow to hate them both.
Some fanfictions have Naruto uncovering dark secrets about Konoha. These secrets could be about the actions of the previous Hokages or the village elders. Discovering these secrets makes Naruto lose faith in Konoha and start to hate it.
It completely flips the original story. In the original, Naruto loves Konoha and eventually comes to terms with his family legacy. But in this fanfic, the hatred changes his character motivation. Instead of striving to protect Konoha, he might be out to get revenge or simply distance himself from it.
In a fanfic, Naruto's bloodline could be connected to an ancient prophecy that Konoha tried to prevent from coming true. They feared Naruto's potential power from the bloodline. So they isolated him and treated him badly. As he grows up and learns about this prophecy and Konoha's actions to stop it, he starts to hate the village. His bloodline could give him unique abilities like the ability to control a powerful and ancient energy source that Konoha was afraid would disrupt their power structure.
In such a fanfic, Naruto might find that his bloodline unlocks a new form of chakra. For example, his normal blue chakra could transform into a more powerful and unique - colored chakra, like purple. This new chakra could have different properties, such as being able to break through any jutsu defenses more easily. As he leaves Konoha, he has to quickly learn to control this new power on his own, without the guidance of his former teachers.