Often in fanfiction, Naruto ignores his family when he becomes obsessed with power. He may think that the power he gains is more important than his family ties. For example, if he discovers a new form of chakra or a new jutsu, he might seclude himself to master it, ignoring his family's concerns. This can lead to his family feeling neglected and trying to bring him back to his senses.
In some fanfictions, Naruto ignoring his family can be a plot device to create drama. It might be due to misunderstandings. For example, if there's a miscommunication about his ninja duties and family expectations. This can lead to a more complex and angsty storyline as Naruto has to face the consequences of his actions, like strained relationships with his family members.
The plot might involve Naruto being influenced by some external force. For example, a new enemy could be using some kind of mind - control or illusion on him to make him ignore the Rookie 9. Another possibility is that Naruto has become so focused on his training to become stronger that he starts to neglect his relationships with the Rookie 9, thinking that his power is more important for the future of the village.
It could be Orochimaru. He's always been interested in Naruto and might come up with a plan to drive a wedge between Naruto and the Rookie 9. By making Naruto ignore them, he could be trying to isolate Naruto and make him more vulnerable to his attacks or manipulations.
In some fanfics, Naruto may ignore Sakura because he has his own emotional traumas to deal with. He might find Sakura's presence a reminder of things he wants to forget or move past, so he chooses to keep his distance. For example, if he has faced a lot of rejection from her in the past, he could decide to just cut her out of his life.
It could be the person who spreads false rumors about Jiraiya to Naruto. This person might be an enemy of Jiraiya who wants to drive a wedge between him and Naruto. By making Naruto ignore Jiraiya, they hope to weaken both of them.
There is a fanfic titled 'Naruto's Solo Journey' where Naruto decides to distance himself from Sakura. He believes that she is too focused on Sasuke and he wants to carve his own path without getting involved in their drama. This makes for an interesting read as it shows Naruto's growth in a different light.
Sakura might initially be confused. She would probably wonder why Naruto is acting this way all of a sudden. For example, she might try to get his attention by calling his name a few times, but when he doesn't respond, she could be left feeling a bit bewildered.
In this kind of fanfiction, Harry might face various consequences. He could end up in dangerous situations all alone without the support and guidance of the Order. Maybe he stumbles upon a powerful dark magic artifact while on his own and has to figure out how to deal with it without the proper knowledge from the Order.
Another reason could be that Naruto is trying to protect Sakura in his own way. He might think that by ignoring her, he can keep her out of harm's way. If he senses that there are dangerous enemies lurking around, he could be trying to create a distance between them so that Sakura doesn't get caught up in any potential battles or threats.