The village's morale could take a huge hit. Naruto is a symbol of hope and strength. If he jumps off the monument, it would make the villagers question their own safety and the stability of the village. Politically, it could cause unrest as the village leaders would have to deal with the aftermath and try to keep the situation under control.
He could get seriously injured. The Hokage monument is very tall, and the fall would be a long one. Naruto might break his bones or worse.
In some fanfiction scenarios, Naruto might jump off the Hokage monument to follow someone or something. For example, if there is a mysterious figure or a spiritual entity that he believes he needs to follow, and the only way seems to be by jumping. It could also be a symbolic act of breaking free from the expectations that come with being associated with the Hokage monument.
He would likely be seen as a traitor by the village. The villagers, who once saw Naruto as a hero - in - the - making, would now turn against him. He'd be hunted by the ninja forces of Konoha.
Maybe Kakashi. He's been Naruto's teacher and mentor. He has the skills and the loyalty to Naruto. He could use his Lightning Blade technique to quickly reach Naruto during the fall and carry him to safety.
A major consequence is how other characters react. The village elders might treat him differently, perhaps with more reverence or more suspicion. Also, his enemies might see him as an even greater threat. Naruto himself might also feel a greater sense of responsibility towards the village, as he is now connected to its founder in a very special way. This could lead to him making different decisions in battles and in the overall story arc.
If Naruto changes something major in the past, it could disrupt the timeline. For example, if he prevents the Uchiha massacre, Sasuke's path would be completely different. He might not have the same drive for revenge, which could mean he doesn't leave Konoha. This would change Naruto's own growth as a ninja, as Sasuke was a big part of his motivation to get stronger.
In fanfiction, Naruto's rule as the 5th Hokage might be full of positive changes. He would probably focus on improving relationships between the different ninja clans in the village. He'd also likely invest a lot in training the new generation of ninjas, using his own experiences to teach them about perseverance and the importance of friendship.
Sakura would be extremely distraught. She has always cared deeply for Naruto, and seeing him in such a dangerous situation would make her rush to his aid immediately. She might use her medical ninjutsu to heal him if he's injured. Kakashi, on the other hand, would be shocked but also proud that Naruto has the courage to face his inner demons in such a drastic way. He would try to understand Naruto's motives and help him get back on track.
He would probably use his leadership skills. Naruto has grown into a great leader as Hokage. In the past, he could quickly assess situations and lead others, whether it's in a battle or in village - building activities.
Under Naruto as the 5th Hokage in fanfiction, the village would experience a cultural shift. Since Naruto values all kinds of people, the village would be more open to different ideas and ways of life. New forms of ninja techniques might be developed as he encourages everyone to explore their potential. Also, the village's defenses would be strengthened not just by physical means but also by the strong bonds between the villagers, which Naruto would foster.