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.
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.
For the other shinobi in the Hidden Leaf Village, Gaara's protectiveness towards Naruto could change their perception of Gaara. They might start to trust him more and see him as an ally rather than just a former enemy. This could lead to more cooperation between the Sand Village and the Leaf Village. For the younger generation of shinobi, like Boruto and his friends, they could look up to Gaara as an example of loyalty and brotherhood. Gaara's actions towards Naruto could inspire them to be more protective of their own friends and comrades.
In a Naruto - is - Gaara's - older - brother fanfiction, the story could start with Naruto finding out about their relation through some ancient scroll or a prophecy. Naruto would then immediately set out to find Gaara. Once they meet, there could be initial tension as Gaara is used to being alone. But Naruto, with his big - hearted nature, would slowly break down Gaara's walls. He could help Gaara with his sand control by sharing his own experiences with chakra control. They might also face a common enemy that forces them to rely on each other and strengthen their brotherly bond.
Sakura might be a bit annoyed at first. She's used to Naruto being independent and always charging forward, and Gaara's overprotectiveness could seem like it's getting in the way. Kakashi, on the other hand, would probably be more understanding. He'd see the value in Naruto having someone looking out for him, but he might also try to encourage Gaara to let Naruto grow on his own a bit. Sasuke could have a very negative reaction. He might see it as Gaara trying to steal Naruto away from him in a sense, especially if he has his own unresolved feelings towards Naruto.
One possible story could be that Sakura goes to the Sand Village for a medical mission while Gaara is Kazekage. They start to interact more, and Gaara is impressed by Sakura's strength and kindness. Eventually, they form a strong friendship that helps in strengthening the bond between the Leaf and Sand villages.
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.
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 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.