Yes, there might be such fanfiction in the vast world of fan-created works. There are many creative fans out there who like to combine different fictional universes in unique ways. Harry Potter and Gaara are from two very different fictional settings, and some fans enjoy exploring the idea of family relationships between characters from different stories.
One possible plot could be that there is a powerful, ancient magic that links the two universes. Harry Potter and Gaara are unknowingly brothers due to this magic. A dark entity threatens both worlds, and only by combining their powers and the knowledge of their two different cultures can they defeat it. Along the way, they have to overcome their differences in personality and upbringing. Harry, who grew up in the wizarding world, has to understand Gaara's more harsh and lonely upbringing in the Hidden Sand Village. And Gaara has to adapt to the more complex social and magical structure of the Harry Potter world. They would form a bond stronger than just brotherhood as they fight side by side against the common enemy.
Their relationship would change in many ways. Firstly, Naruto would be more involved in Gaara's life. He would probably train with Gaara more often, teaching him new techniques. Secondly, Naruto would be a source of emotional support for Gaara. Given Naruto's positive attitude, he could help Gaara overcome his past traumas more effectively. And finally, they would have a stronger sense of loyalty towards each other. In battles, Naruto would not hesitate to put himself in harm's way to protect Gaara, and Gaara would do the same for Naruto, much like a real - life brotherly relationship.
Naruto is often paired with Gaara in these fanfictions. Their strong bond in the original series makes it interesting to see how they interact when Gaara time - travels. Naruto could be the one to help Gaara adjust to a new time period or vice versa.
Often, Naruto's female characters are paired with Gaara. For example, Hinata. Since Hinata is a kind - hearted and shy character, her relationship with the initially cold - hearted Gaara could be an interesting exploration of how two different personalities come together.
Well, it could change the storyline in many ways. For one, Naruto would have an even stronger support system. Gaara, being overprotective, might prevent Naruto from getting into some of the more dangerous situations he usually jumps into. This could potentially slow down Naruto's growth in terms of combat experience as he'd be shielded from some threats. But on the other hand, it could also give Naruto more emotional stability, which could in turn help him better control his chakra and learn new jutsus faster. Gaara's presence as an overprotective brother could also affect how the other characters interact with Naruto. Some might be more cautious around him because they know Gaara is watching over him like a hawk.
Temari and Kankuro, Gaara's siblings, are likely to be in the fanfiction as well. They could play protective roles towards their little sister, or they might have some conflicts with her as she tries to assert her own identity. And there could be some antagonists, like rogue ninjas from other villages who see Gaara's sister as a target for some reason.
One plot idea could be that Gaara, as Naruto's protective brother, uses his sand powers to shield Naruto from various attacks during a major battle against a new and powerful enemy. For example, when the enemy launches a surprise attack on Naruto, Gaara quickly forms a sand barrier around him, saving his life. Another idea is that Gaara helps Naruto train in secret to master a new technique. He shares his own experiences with controlling power, which Naruto can relate to with his Nine - Tails chakra.
Naruto could still be a main character. Their relationship would likely be different if Gaara were a girl. Maybe Naruto would be more protective of her.
It could be Orochimaru. He might manipulate Gaara into kidnapping Sakura. Orochimaru could have his own devious plans for Sakura, like using her medical knowledge for his experiments. So, he uses Gaara's power and his somewhat unstable mental state at times to achieve his goal. Then Sakura and Gaara would have to team up to fight against Orochimaru.