There are likely fanfictions where Naruto's immortality in the Fairy Tail setting gives him unique powers. For example, he might be able to combine his ninja skills with Fairy Tail magic in ways that no one else can. His immortality could also mean he has an infinite amount of time to master different types of magic, making him an extremely powerful and feared figure in the Fairy Tail world.
I'm not sure of a specific one off - hand, but it could be a fanfiction where Naruto's tail is a key to unlocking a hidden dimension. In this dimension, there are new forms of chakra and powerful jutsu that he can learn. The story could follow his journey to discover the secrets of this dimension with the help of his tail.
In fanfiction, anything can happen. The author might create a crossover scenario where Naruto somehow gets the Sharingan and ends up in the Fairy Tail universe. It could be due to some magical or scientific experiment gone wrong or right, depending on how the story is framed.
There might be a fanfiction where Naruto's tail is a result of some new form of chakra transformation. In this story, his tail could be a sign of a new power he's developing. For example, it could be related to some ancient and forgotten ninja art that gives him this tail, and he has to learn how to control it and use it to his advantage. He might face challenges from other ninja who are either jealous or afraid of this new power.
It would be a wild change. Naruto's Rinnegan powers would make him a major force in the Fairy Tail world. He might be able to use the Preta Path to absorb magic attacks, which would be a huge advantage. And his Deva Path could manipulate gravity in ways that are unheard of in Fairy Tail. This could lead to new alliances or rivalries. Maybe some Fairy Tail members would be in awe of his power and want to learn from him, while others might be jealous or afraid.
Maybe Naruto gets these wolf ears and tail as a result of a powerful jutsu gone wrong. This could lead to a comical yet heartwarming story where he has to learn to control his new appendages. For example, his tail might wag involuntarily when he's excited or his ears might twitch when he senses danger. His friends could help him adjust and in the end, it could become an advantage in battles.
In some fanfictions, Naruto might get wings and a tail due to a powerful curse or a strange experiment gone wrong. Maybe an enemy used some forbidden jutsu on him that altered his physical form.
One possible plot could be that Naruto, with his fox tail which might be a sign of his special power or heritage, somehow ends up in the world of Fairy Tail where he meets Erza. They start off as strangers but gradually get to know each other through various battles and challenges. Naruto's unique abilities, including those related to his fox tail, intrigue Erza and they start to develop feelings for each other as they fight side by side against common enemies.
One interesting storyline could be Naruto using the Sharingan to copy the magic of Fairy Tail wizards. He could then combine his ninja skills with the copied magic, creating unique fighting styles. For example, he might copy Erza's requip magic and use it to quickly change his ninja tools during battle.
The Sharingan in such a fanfiction could be used as a tool for a Fairy Tail character to gain an edge in their battles. They might initially struggle to control it, but as they learn more about it, they become a formidable force. For instance, they could use the Sharingan's genjutsu abilities to deceive opponents in the Fairy Tail battles.