One possibility could be that Harry is so traumatized by the war that he wants to isolate himself from the wizarding world. He might go into hiding in a remote, Muggle - free location to deal with his inner demons.
His family, if he has any remaining family members at this point, would also be deeply affected. They would be worried sick about his whereabouts and what could have happened to him. And any young witches and wizards who looked up to Harry as a hero would be left confused and disheartened by his sudden disappearance.
Often, his friends in Konoha will start to search for him. They might split into teams and scour the surrounding areas. For example, Sasuke could use his Sharingan to look for any signs of Naruto's departure. Meanwhile, back in Konoha, there could be unrest as the villagers start to worry about the safety of the village without Naruto.
One possible reason could be for the sake of creating a dramatic plot. The disappearance of Naruto might be used to show how the female characters respond to his absence, highlighting their growth and independence. For example, Sakura might take on more leadership roles in the village without Naruto around.
One possibility could be that Naruto is kidnapped by a hidden enemy right at birth. Maybe a rogue ninja group has a vendetta against the Fourth Hokage and they steal Naruto away. This would lead to a completely different story arc in the Naruto universe, with Konoha perhaps not knowing Naruto's true identity and not treating him as the Jinchuriki. It could also mean that Naruto grows up in a different environment, away from the discrimination he faced in the original story, but also without the friends and support system he eventually had in Konoha.
Orochimaru could be a likely main villain. He has always been interested in Naruto's power and body. Maybe he finally devises a plan to kidnap Naruto and use his body for his own reincarnation experiments.
It could be that Naruto sacrifices himself to seal away a great evil. However, his friends don't believe he's truly gone and embark on a journey filled with self - discovery and growth as they try to find a way to bring him back. Along the way, they might encounter other lost souls or forgotten powers that could potentially help in their quest.
In some fanfictions, Naruto becomes more jaded after the war. He has seen so much death and destruction that he loses some of his initial idealism. He might be more cautious in his actions and less quick to trust.
In some fanfictions, Naruto and Hinata start a family right away. They enjoy a peaceful life in Konoha, with Naruto being a great Hokage and Hinata supporting him at home and also taking part in some village activities.
One common theme is rebuilding. After the war, the villages, especially Konoha, need to be rebuilt both physically and socially. Another theme is reconciliation, like Sasuke coming back to Konoha and being accepted again. Healing is also a theme, as the characters have to deal with the mental and physical scars of the war.