In terms of his relationships with other Jounin, there would be a shift. At first, some might be skeptical of him. But as he proves himself on missions, they would start to respect him. He would be invited to more Jounin - only meetings and events. His relationships would expand to include more political and strategic aspects as he becomes part of the higher - echelon of Konoha's ninja force.
Naruto's relationship with his students, if he has any as a Jounin, would be very important. He would be a mentor figure. He would have to be strict but also kind, just like Kakashi was to him. He would pass on his values and ninja techniques. His relationship with the Hokage would also be different. He would be closer to being an equal, having more say in important decisions regarding the village's security and future.
His relationship with his old friends like Sasuke would likely become more of an equal partnership. Since they are both highly skilled, they could collaborate on important missions. With Sakura, he might be seen as more of a respected colleague, and she could look up to him in a different way as a Jounin.
If Naruto becomes a Jounin early in fanfiction, his relationship with his peers might change. They could look up to him more, seeing him as a role model. His friends like Sasuke might be a bit jealous at first but would also be proud. His relationship with his teachers could shift as he is now closer to their rank.
Naruto's relationship with the younger generation would be very different. As a Jounin, he would be a role model. The kids in the village would look up to him even more. He could inspire them to work hard. His relationship with Hinata might also evolve. Maybe he would finally have more time to focus on their relationship now that he has achieved a higher rank and more stability in his ninja career.
In terms of his romantic relationships, if there was a budding relationship with Hinata, for instance, his new status as a jounin could either strengthen their bond as she admires his achievement, or it could create some distance as he has more duties and less time for personal matters.
His friends in Team 7, Kakashi included, would be deeply concerned. They would try to figure out what made Naruto change. However, Naruto's coldness would make communication difficult. He might distance himself from them, seeing their attempts as interference. His relationships with the villagers would also change. People who once looked up to him might now be afraid of him, as his cold demeanor is a far cry from the hero they knew.
Well, Naruto's relationships would change significantly. For instance, his relationship with his sensei, Kakashi, could shift. Kakashi would become more of a mentor in a different way, guiding Naruto through this difficult time. With his peers, they would no longer see him as just the powerful ninja he once was. Some might pity him at first, but Naruto's positive attitude could turn that pity into respect. And his relationship with the village as a whole would change too. He would become more of an inspiration, a symbol of perseverance in the face of adversity.
His relationship with Sasuke would be very different. Sasuke might see Naruto as a threat at first, given the power of the Juubi. But eventually, they could come to an understanding. Naruto would still want to protect Sasuke, and Sasuke might realize that Naruto's intentions are pure even in his Juubi form.
His relationship with Kurama might become more complex. Since he's now also a kitsune - like being. They could either become more competitive or find a new way to cooperate. With his friends, some might be in awe of his new form, while others might be a bit scared at first. But Naruto's kind heart would eventually make them all accept him as before.
If Naruto becomes a nukenin, his relationship with the village elders would be completely severed. They would see him as a threat, and he would see them as unjust. With his peers, some might side with him, believing in his innocence. Others would follow the village's orders out of loyalty. His bond with the Nine - Tails would also change. The Nine - Tails might offer more support, seeing Naruto as an outcast like itself.
His relationship with Sasuke would be greatly affected. Sasuke, who has his own experiences with the idea of killing, might see Naruto in a new light. They could either become rivals again or Sasuke might try to 'save' Naruto from his new dark path. With Sakura, she would be horrified. She's always seen Naruto as a positive force, and this change would make her question everything. She might even try to use her medical skills to 'cure' Naruto of his new - found killer tendencies.