His decision to leave the Dursleys can make the plot deviate in many ways. In fanfic, it could mean that he doesn't have to wait for the official Hogwarts invitation. He might seek out the wizarding world on his own. This could lead to him encountering different characters or facing different challenges earlier. Also, his departure might make the Dursleys more or less relevant in the overall story. If they are no longer a factor, then new antagonists or plot elements need to be introduced to keep the story interesting.
It could go either way. If the fanfic shows Harry in a really positive light, like a super - powerful wizard who is also kind - hearted despite their treatment of him, they might feel a bit guilty. But if the fanfic portrays Harry in a negative way, they might feel justified in their own cruel behavior towards him.
The story could take a darker turn if he refuses to go back to the Dursleys. Maybe without the protection of the blood magic at the Dursleys' place (even if they were cruel), he becomes more vulnerable to Voldemort's attacks earlier. Or, on the other hand, he could find a new form of protection that was not part of the original story, which could change the power dynamics in the fight against Voldemort.
Well, it could completely disrupt the relationships. For instance, his friendship with Ron and Hermione might be strained as they try to convince him to stay. This is different from the original where they are all united in their Hogwarts experience.
In some 'Harry Potter and the Dursleys' fanfics, there's a plot twist where the Dursleys are actually protecting Harry from a greater threat that the wizarding world doesn't even know about. They've been cruel to keep him close and safe. Also, there could be a twist where Vernon Dursley has a connection to the wizarding world through his family's past that he's been trying to hide all along, which changes how we view his hatred for magic.
Perhaps he gets a chance to attend a special magical school or training program that requires him to leave the Dursleys. It could be a place where he can develop his powers without the Dursleys' interference. Also, he may be running away from a threat that the Dursleys are unable or unwilling to protect him from.
One possible plot development could be that Harry discovers his magic earlier on his own due to the neglect. He might start exploring the wizarding world secretly, finding hidden magical artifacts in the neighborhood. Another could be that he forms an unexpected alliance with the local magical creatures that sense his isolation. Maybe he even starts teaching himself basic spells from an old, dusty spellbook he finds in the attic.
The abuse makes Harry more cautious. He has learned from the Dursleys that the world can be cruel, so he's careful about who he trusts. For example, he might be hesitant to open up to new people he meets at Hogwarts.
They are often portrayed as the villains. Their years of treating Harry badly are brought to light. They might look scared and out of place in the wizarding courtroom, with all its strange sights and magic.
In this fanfic, the Dursleys may be depicted as being in over their heads. Vernon Dursley, for example, might be shown as constantly angry and trying to assert his authority in a situation where he has none. Petunia Dursley could be portrayed as being torn between her dislike for magic and her sense of family obligation towards Harry. Dudley might be shown as being bullied or affected in the Muggle world because of Harry's presence in their family.
Harry's character might grow in resilience. The neglect forces him to become self - reliant. He has to fend for himself, which makes him tougher. For example, he learns to cook his own meals and take care of his basic needs without any help. This self - sufficiency builds his confidence over time.