Yu Yu Hakusho remains a popular and well-received series. However, there were still some story threads left unresolved by the end -- primarily due to Yoshihiro Togashi ending the series for the sake of his physical and mental wellbeing. The most significant conflict that fans never got overtly resolved was the secret identity of Hiei as Yukina's brother. While Hiei kept this truth from his sister, much to some fans' dismay, his decision may have been for the best -- or was it?
Hina -- Yukina and Hiei's mother -- was from the Glacial Village, home of the Ice Apparitions. While Hina used a male to conceive children, such an act went against the village's beliefs. Since Hiei was a male and a fire demon, he was labeled dangerous by the village elders. He was thrown from a cliff by his mother's best friend Rui, and Hina died shortly after, with the reason being suicide in the anime and no reason given in the manga. He grew up hating the Ice Apparitions for this and created a self-fulfilling prophecy. Meanwhile, Rui raised Yukina and told her about her twin brother, and Yukina set out to find him.
After Hiei, Kuwabara and Yusuke rescue Yukina from Guro Tarukane, who was using her to create crystals, Hiei only admits to being a half-sibling to her rather than the twin she's been searching for -- claiming they have different mothers. While Hiei's backstory is delved into more in-depth towards the end of the series, Yukina never learns the truth. At least, not overtly. There are hints that she suspects Hiei is her brother in the OVA Eizou Hakusho, mostly when Hiei tries to return his mother's tear to Yukina, but she insists that he keep it.
The OVA never aired in the West with the rest of the series, so unless fans knew about it, they wouldn't get the closure it offers regarding this significant plot point. Still, that leads to the question of why Hiei never just admitted to being her brother in the first place and whether it was for the best. He had the opportunity to tell her after he helped rescue her, but instead, he chose to lie about it. His attitude changes when he's around her, becoming more protective and sincere, though only towards her.
One explanation for why Hiei never told Yukina, or at least a partial motivator for keeping it secret, was the promise he made with Shigure. When receiving his Jigan Eye from Shigure, the surgeon had Hiei swear never to tell Yukina that he was her brother. However, it's never entirely clear why Shigure would have Hiei make such a promise. Nonetheless, Hiei accepts, revealing that he wasn't planning to tell Yukina the truth anyway, just that he would find and protect her.
Valient as Hiei's intentions are, his plan completely neglects Yukina's feelings. Even if he had found her before she was captured, how would he have approached the situation? Yukina is pretty relentless in her mission of finding her brother, so it's unlikely she would have stopped searching, so she'd be putting herself in more danger.
Both humans and demons highly prize Ice Maiden tears, so having Yukina search the world for the brother right next to her would have done nothing but cause more problems. Not only that, but Hiei stole from Spirit World, becoming a felon in the process. Had he not been placed on parole in the Human World and instead more harshly punished, the two may have never been reunited in the first place, and Yukina's rescue may not have happened.
Hiei's primary motivation for keeping his secret is that he doesn't want to tarnish the image Yukina has of her brother, or rather the one he thinks she has. It's never clearly stated by Yukina what she thinks her brother is like, so Hiei places his insecurities and fears about the truth onto her without thinking about how it made her feel. While he would have been protecting her physically, her feelings mattered just as much as his. Everyone else in the group, save for Kuwabara, knows, so it doesn't seem kind to withhold the truth from Yukina. In the end, the only person Hiei was protecting was himself.