In a world where most of the characters become soldiers at a young age and deal with constant warfare, it's no surprise that some of them are dealing with mental health issues - especially since this world fails to provide its soldiers with any mental health care, even when they've experienced something as severe as having their entire family slaughtered by their older brother. Though no one in Naruto has been diagnosed with anything, many of the characters are in desperate need of some support.
Reddit user u/Sarahmint had a fascinating fan theory that describes how different characters from Naruto represent different mental health issues. This list showcases some of their ideas and brings in ideas about mental illness in Naruto from fans all over the Internet. These theories aren't necessarily the cold hard facts about the cast of Naruto - rather, they're a lens through which to interpret the characters, and perhaps come to a fuller and richer understanding about who they are and how they interact with their story.
u/Sarahmint believes that the Nine Tails residing in Naruto's body is actually a representation of mental illness. The Nine Tails is a beast that lives inside of him, which he must get control over if he wants to live a peaceful life. This is analogous to having a mental illness that creates chaos in one's personal life and getting appropriate treatment for it. But Naruto does more than just control it - he also finds ways to work with the Nine Tails and use its strength to improve his own life. If the Nine Tails is a mental illness, the message of Naruto is that mental illness isn't all bad - it just takes a lot of work to get to that point.
It's possible to read Gaara's story as the inverse of Naruto's. If the tailed beast sealed into a person's body is a symbol of latent mental illness, then Gaara has one, too. Gaara's village utterly rejects him and sees him as a violent monster when he has no desire to harm anyone. After a series of traumatic events, he starts to see himself as the monster that everyone says he is and takes out his pain and anger on his village.
People with mental illnesses are often expected to be violent and are treated poorly as part of that stigma. It's common for people to eventually start acting on the expectations of others, which is what happens in Gaara's case.
Reddit user u/paperclip1213 believes that Karin Uzumaki of Team Taka may have a borderline personality disorder, also known as BPD. People with BPD fear being abandoned and can take extreme measures to avoid this - even hurting themselves or other people. They tend to see everything in extremes - either they love someone and think that they can do no wrong, or they utterly loathe them - and they can change their minds quickly. Impulsive behavior, mood swings, and inappropriate anger can also come with the condition.
Karin displays most of these symptoms in her interaction with others. She adores Sasuke and is willing to let him bite her arm in order to heal himself, even though he isn't even basically civil with her. She becomes furious with Suigetsu over light teasing or perceived slights. Her mood cycles rapidly depending on the circumstances.
Is Hinata Hyuuga just shy, or does she actually suffer from social anxiety? People with social anxiety fear interacting with others because they fear being judged harshly. This can harm their self-esteem, create feelings of self-consciousness and embarrassment, and can even cause depression. Raised in a strict family where she felt she could never measure up to their expectations no matter how hard she tries, Hinata fears interacting with others because of how they might respond to her. She has a low opinion of herself and gets incredibly embarrassed whenever she has to speak to other people - particularly those who she wants to impress.