One plot implication could be a shift in the power balance in the Naruto universe. With Naruto having the Rinnegan, the Akatsuki or other villainous groups might see him as an even greater threat and target him more aggressively. Another aspect could be the relationship between Naruto and Sasuke. Sasuke giving away such a powerful ability shows a great deal of trust and sacrifice. It could lead to a deeper bond between them, and they might team up more often to face new challenges.
One possible plot could be Naruto feeling a sense of loss and vulnerability without his headband. Maybe he loses it during a battle and has to prove himself all over again without that symbol of his ninja identity. He could struggle with how others in the village view him now that he doesn't have the recognizable headband.
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.
Well, if Sasuke gives Naruto his Rinnegan in fanfiction, Naruto's abilities would be greatly enhanced. The Rinnegan is extremely powerful, granting abilities like the Six Paths techniques. Naruto might gain better control over space - time, be able to use powerful gravitational and repulsive forces more effectively, and perhaps even access new forms of chakra manipulation. This could make him an even more formidable opponent, able to take on stronger enemies with ease.
The story could progress in many ways. For one, Sasuke might be filled with guilt for receiving Naruto's eyes. This guilt could drive him to become a better person and fight for the things Naruto believes in. He may also use the new vision to uncover secrets or powers that were previously hidden. On the other hand, Naruto, without his eyes, could develop other enhanced senses or abilities, and together they could form a new kind of partnership. They could also face challenges from those who oppose the idea of Naruto sacrificing so much for Sasuke.
One interesting fanfiction could be where Naruto slashes his headband as a sign of breaking free from the traditional shinobi ways that have restricted him. Maybe he decides to form his own path, independent of the Hidden Leaf Village's old rules.
His relationship with Sasuke might change. Sasuke might initially be confused or even critical of Naruto without his headband. But as Naruto shows his determination without it, Sasuke could gain a new respect for him. With his friends like Sakura, they might be more protective of Naruto, fearing that others will target him without the symbol of his ninja status.
Maybe Naruto gets tired of always chasing after Sasuke. He has his own life and dreams too. He might think that he has sacrificed enough of his time and energy on Sasuke, and it's time to focus on himself and the village. He could start to believe that Sasuke has made his own choices and he can't keep forcing Sasuke to come back.
One possible reason could be their deep bond. Naruto and Sasuke have a complex relationship. Naruto has always wanted to save Sasuke from the darkness. Giving his eyes might be the ultimate way to show his loyalty and belief in Sasuke's redemption. It could also be because Naruto sees Sasuke as a part of himself, and sacrificing his eyes is like sacrificing for a part of his own soul.
If Naruto gives up on Sasuke in fanfiction, the storyline could take a very different turn. For one, the whole dynamic of their relationship, which has been a central part of the original Naruto story, would be completely disrupted. Naruto has always been determined to bring Sasuke back to the village. Without that drive from Naruto, Sasuke might become even more isolated and perhaps fall deeper into the darkness. This could also lead to different power alignments in the ninja world. Allies of Naruto might question his change of heart, and enemies could take advantage of the rift between the two former friends.