He might be very disappointed at first. Naruto had always dreamed of becoming Hokage, so it would be a huge blow. But he could also use this as motivation to prove himself even more. He would likely go on some self - improvement journey, training harder and showing the village that they made a mistake.
One possible reason could be that some elders in the village still have doubts about his past. For example, his connection with the Nine - Tails. They might fear that this could lead to instability in the village. Another reason could be political intrigue. There might be other powerful clans or individuals who want the position for themselves and manipulate things behind the scenes to deny Naruto.
Naruto could be confused. He may not understand why he was denied when he has accomplished so much. He might start to question the system and those in power. This could lead him to have some internal conflict as he tries to figure out whether he should continue to strive for the promotion within the existing framework or find his own path.
He might feel angry and frustrated. Naruto has always been proud of his potential and his roots in some way. Being denied his heritage would be like taking away a part of his identity, so he'd likely be quite mad at those who are doing the denying.
Naruto might initially be very disappointed. He has always been eager to train and become stronger. So being denied by Kakashi could make him feel a bit down. But he could also be defiant. He might start training on his own, determined to show Kakashi that he doesn't need his approval to get stronger.
One idea could be a fanfic where Danzo actually manages to become Hokage through some underhanded means. Then Naruto has to find a way to expose his true nature and restore the rightful leadership to the village. Maybe he teams up with some of the other shinobi who were also suspicious of Danzo's actions all along.
It would be a huge blow to his character. Naruto's whole identity is somewhat tied to becoming Hokage. If he's denied, he could go through a period of self - doubt. However, this could also lead to him exploring other aspects of his life. Maybe he starts to focus more on teaching the next generation of ninja, passing on his skills and values even without the Hokage title, which in turn could give him a new sense of purpose.
One possible interesting story could be Naruto facing some sort of political conspiracy within the village that leads to his hokage denial. Maybe some elders or a hidden group want someone else in power for their own motives.
The other Konoha 11 members would likely be involved too. They might be split in their opinions. Some would firmly stand by Naruto, while others might be swayed by the village's reasons for denying him. Hinata, for instance, would probably be one of Naruto's biggest supporters and might even try to fight against the decision in her own way.
Perhaps there are external threats that the village is facing. The leaders might think that Naruto is more valuable in his current position, where he can be used as a front - line fighter. They may believe that promoting him would mean taking him away from the direct combat and strategic roles that he currently excels at, and they can't afford to do that during a time of potential danger.
It could be due to the current situation in the village. There might be threats that required Naruto to stay. Tsunade, as the Hokage, had to consider the overall safety and stability of the village. So, she denied Naruto's request for the training trip to ensure he was available in case of any emergency.