One possible reason could be fear. Naruto has the Nine - Tails within him, and some in Konoha might be too afraid of its power getting out of control. They may not fully trust Naruto to keep it in check.
At first, Naruto is likely to be heartbroken. Konoha has been his home, and he has always longed for acceptance from the villagers. So being abandoned would be a huge blow to him.
One possible reason could be that Naruto discovers some dark secrets about the village leadership that go against his values. For example, if he finds out that they have been using underhanded means to maintain power.
One possible reason could be feeling betrayed by the village leaders. Maybe they made decisions that Naruto strongly disagreed with, like how they treated certain groups or individuals. If Naruto felt that Konoha was no longer the place that stood for justice and fairness as he once believed, he might abandon it.
Many relationships would be affected. For example, Sasuke's story would take a different turn. He might not have the same motivation to return to Konoha if Naruto is not there. Also, Naruto's other friends like Sakura would be devastated and might set out on a journey to find him. The power dynamics in the ninja world would shift as well, with other villages possibly seeing Konoha as weaker without Naruto.
As for Kakashi, he would be in a difficult position. He has mentored Naruto and also has a strong loyalty to Konoha. He might try to understand Naruto's reasons while also having to deal with the village's reaction. Their relationship could become strained as Kakashi tries to balance his duties to the village and his connection with Naruto.
This fanfiction might make Naruto more of a loner. In the original, he was always surrounded by friends and the support of the village. Abandoning Konoha would force him to rely only on himself and his own powers. His character could become more cynical as he distances himself from the place he once called home.
Well, it could go in many directions. For example, his family might start to search for him, which could lead to a series of adventures where they encounter new characters and face various challenges. His wife, if he has one in the story, might become stronger and more independent in his absence, vowing to bring him back.
Naruto would be devastated. He sees Kakashi as a mentor and father figure. He might initially go into denial, not believing that Kakashi would actually abandon him.
Well, in some fanfictions, Naruto might outsmart Konoha by using his unique knowledge of the village's secrets that others don't know. For example, he could use his understanding of the old tunnels under Konoha to his advantage in a situation where he needs to escape or gain an upper hand in a conflict.