Naruto might end up feeling extremely lonely. His family has always been an important part of his life. Without their support and presence, he could become depressed or start making even more reckless decisions. Also, if his family tries to search for him, they could get into danger themselves in the process, which would be an unintended consequence of his hiding.
In the long run, if he continues to hide from his family in the fanfiction, his relationships with other friends and allies could also be affected. His family might spread the word that he's missing or hiding, and some people might start to distrust him too. Plus, he may miss out on important family events and celebrations, which could create a rift that is hard to mend later.
Perhaps he has hidden the full extent of his Sage Mode abilities. He might be able to stay in Sage Mode for a much longer time than he has shown so far, or have some other undisclosed powers within that mode, like the ability to sense enemies from much greater distances.
Well, in fanfictions, often the plot takes a different turn. Naruto might hide from his family because there could be some sort of misunderstanding. For example, there could be false rumors about him being involved in something bad. And he might think that his family would believe those rumors without giving him a chance to explain. So, to avoid the immediate confrontation and judgment, he chooses to hide.
One consequence could be that his friends might start to suspect him. Since Naruto is usually an open book, hiding something as significant as his eye could make them wonder what he's up to.
Naruto could be afraid of being seen as weak. In the ninja world, weakness is often frowned upon. He has a reputation to uphold, especially in front of his family. He doesn't want them to think that he can't handle his problems, so he hides his pain. He also might be trying to be strong for his family. If they see him in pain, it could affect their morale. So, he pretends that everything is okay to keep their spirits high and to give them the impression that they can rely on him no matter what.
His friends might react with shock and confusion. They might not understand why he has been hiding this from them, and it could cause a rift in their relationships until he explains the reasons behind it. For example, some friends might feel betrayed that he didn't trust them enough to share this secret.
It could also lead to a divergence from the core values of the Naruto series. The series often emphasizes family, friendship, and love in a positive way. If Naruto hates his family in fanfiction, it goes against these values and may cause some fans to be disappointed.
He might put on a fake smile and act normal. Naruto is good at being energetic on the outside, so his family may not suspect that he's in pain.
He might want to protect himself. If others know how smart he is, he could become a target. Enemies might see him as a greater threat and go after him more aggressively.
One possible consequence is that he could lose the trust of his friends. If everyone believes he is responsible for something his sister did, his friends might start to distance themselves from him.
One possible consequence could be a change in his fighting style. Without his eye, he might have to rely more on his other senses like hearing and smell, similar to how blind ninja in the Naruto universe adapt. For example, he could focus more on using his chakra to sense the movements of his opponents.