Naruto's relationship with his enemies would surely be affected. Enemies might target him more if they think his red hair is a sign of a new power they can steal or a weakness they can exploit. On the other hand, some enemies might be more hesitant to attack him, fearing that the red hair is a sign of a new and unknown strength. His relationships within the ninja world in general would be in flux, as everyone tries to figure out what this red - haired Naruto means for the balance of power.
In terms of his relationship with the villagers, those who once bullied him for being an orphan and having a different look (with his blond hair) might be even more hostile if his red hair makes him stand out more. However, there could also be some villagers who see his red hair as a sign of something special and start to respect him more, which could lead to a shift in how Naruto views the village and his place in it.
His relationship with Sakura might change. She could be more intrigued by his new look and start to see him in a different light, not just as the annoying kid. With Sasuke, Sasuke might be more jealous as Naruto's blue hair makes him even more unique and gets more attention.
In some fanfics, his long red hair might make him more attractive to certain characters. For example, Sakura could start seeing him in a new light, as the change in his appearance makes him stand out more. It could also make his enemies more cautious, thinking he has some new power associated with the hair.
His friends might be initially shocked. Sakura could tease him about it at first but then grow to like the new look. Sasuke might be indifferent on the surface but secretly find it interesting. His enemies might underestimate him, thinking he's become more of a softie with the long hair, but then be surprised by his unchanged combat skills.
His relationship with Sasuke would likely become more complex. Sasuke might be both jealous and intrigued by Naruto's new powers. Naruto would have to prove that he still values their friendship despite his increased strength. With Sakura, she might see Naruto in a new light, perhaps with a bit more admiration as he now has such rare and powerful abilities.
His long hair might make Sakura see him in a different light. Maybe she starts to notice his more mature side with the long hair and becomes more attracted to him. It could also change his relationship with Sasuke. Sasuke might initially mock his new look but later come to respect the power that seems to be associated with Naruto's long hair.
His relationship with Kakashi would also be affected. Kakashi would have to mentor Naruto in a different way. He would need to teach Naruto how to use the Byakugan effectively, which is different from what he's used to with Naruto's normal abilities. This could strengthen their bond as teacher - student, or it could create some friction if Naruto struggles to adapt to Kakashi's new teaching methods.
His relationship with Sasuke would likely be very different. Since Sasuke has also had experiences with the Rinnegan, they might now have a new level of understanding between them. They could train together to master the powers of the Rinnegan or compare notes on how to use its abilities.
It would change it a great deal. His red hair might make him stand out even more, and with the Rinnegan, his power level would be off the charts. Enemies might be more afraid of him, and allies might look at him with a new level of respect or suspicion. The power dynamics in battles would be completely different. For example, he could use the Rinnegan's abilities in ways that no one has seen before, changing the strategies of both his side and the enemy's side in wars and skirmishes.
His friends would be very cautious at first. For example, Sasuke might be more on guard, seeing Naruto as a potential threat. However, true friends like Shikamaru might try to understand what's going on and find a way to help Naruto control his new powers.