The story could take a dark turn. Without his heart, Naruto might become a soulless being, and his friends would be devastated. They might try to find a way to get his heart back. Maybe they search for a powerful sage or some ancient magic that could reverse what has been done.
One possible reason could be a sign of extreme loyalty and friendship. Naruto might see Yang in a dire situation where she needs his arm for some crucial purpose, like to complete a powerful jutsu or to defend against an overwhelming enemy. He is known for his selfless nature, so sacrificing his arm would be in line with his character.
She might become more cautious. Having received Naruto's arm, she realizes the great sacrifice he made. This could make her think twice before engaging in battles recklessly, as she now has a part of someone she respects and doesn't want to waste this 'gift'.
One possible reason could be a series of misunderstandings. Maybe his family made decisions that he felt were against his principles or dreams, and after repeated attempts to reconcile the differences, he just lost hope. For example, if his family was against his ninja way, which is such a core part of his identity.
If Naruto gives up on his friends in a fanfic, it would be a huge departure from his character. Naruto is known for his unwavering loyalty. His friends are his strength. Without them, he might become a much darker and more isolated character. He could lose his drive to become Hokage as his friends were a big part of that motivation. Also, the village that values friendship and bonds might see him in a different light, and he could face a lot of ostracism.
One reason could be the constant setbacks he faces. He might feel that no matter how hard he tries, he can't achieve his goal. For example, if he fails too many times in battles against strong opponents, he could lose hope. Another reason might be the loss of important people in his life. If someone like Iruka or Kakashi, who have been his pillars of support, were to die, it could shatter his dreams as they are so closely tied to his will to succeed for their sake.
One possible reason could be to protect his friends. He might think that by sacrificing his heart, some great danger or enemy would be vanquished, thus ensuring the safety of those he cares about. For example, if there was a powerful curse that could only be broken by his heart's sacrifice.
If Naruto gives up his dream, Sasuke might view him as a weakling. Sasuke has always been competitive with Naruto, and Naruto's giving up could make Sasuke lose respect for him. Their relationship could become strained as Sasuke may distance himself from Naruto, thinking he is no longer the worthy rival he once was.
If Naruto gives up on his dream in a fanfiction, the whole Naruto universe would be drastically different. His friends would be shocked. For example, Sasuke might not have the same motivation to grow stronger as Naruto's unwavering determination was always a driving force for him. The village of Konoha would also lose a great source of inspiration. Naruto was always the one to believe in the village's future and strive for its betterment. Without his dream, the hope he spread might fade away, leading to a more gloomy atmosphere in the village.