It can create a lot of conflict. For example, he might start attacking the village or his former friends. This would lead to battles and power struggles. The other characters would then have to figure out how to deal with him, either by trying to reason with him or by using force to stop his rampage.
It can simplify the plot. Without Naruto's internal struggle with hatred, he may make more rational decisions, which can speed up the resolution of conflicts in the story.
It can create a lot of tension. If Naruto hates Sakura, it changes the dynamic of their team. Their interactions become more strained and filled with conflict, which can drive the plot forward as they have to work together despite their feelings.
Naruto might also face hatred by becoming stronger. He trains harder and shows his opponents that he is not someone to be underestimated. For instance, if a group of rogue ninjas hates him because they think he's too idealistic, he proves them wrong by defeating them in combat and showing that his ideals are what make him strong.
It could also speed up the plot in a different way. If Naruto ignores everyone, some villains might see this as an opportunity to attack. Since Naruto is usually a key defender, his withdrawal could lead to more immediate threats to the village. This would force the other characters to step up in different ways than they normally would. They might have to form new alliances or develop new powers quickly to fill the void left by Naruto's ignoring behavior.
It can strengthen the NaruHina relationship. Since Naruto hates Sakura and Sasuke, he might turn more towards Hinata for emotional support. Hinata's presence becomes even more important to him. He realizes that Hinata is the one who truly cares about him, not like Sakura who is preoccupied with Sasuke.
His friends might start to avoid him. If Naruto is constantly snapping at them, they'll feel hurt and not want to be around him as much.
On the positive side, it could bring him closer to those who also have issues with his father or the old ways. There might be some characters who have been wronged by Naruto's father's decisions in the past. Naruto's hatred could make them form an alliance of sorts, united by their shared negative feelings. However, overall, it would likely cause more problems than benefits in his relationships.
Another way is through his attitude towards the villagers. He would show no respect or kindness to them. Instead, he would be cold and hostile, perhaps even going as far as to physically intimidate those who cross his path in Konoha.
It could lead to a lot of conflict. His friends might try to reason with him, but if he's too angry, he might push them away. This could split their group apart, creating new alliances or just leaving Harry isolated for a while. His anger can also attract the attention of the wrong sort of people, like those who want to use his emotional state for their own gain.
Naruto's hatred could be a distraction. Instead of focusing on his opponents and the tasks at hand, he might be constantly thinking about Kakashi. This could cause him to make mistakes and not perform as well as he could have.