One possible reason could be jealousy. Maybe the sister is getting more attention from their parents or at Hogwarts. For example, if she is a star student and Harry feels overshadowed.
If such fanfiction exists, it may lead to a lot of negative impacts. It can mislead some young and inexperienced readers into thinking that abnormal relationships are acceptable. Moreover, it might face legal issues in some cases as it violates ethical and moral boundaries. Also, it will be shunned by the vast majority of the legitimate Harry Potter fandom who uphold the positive values of the series.
Their parents' old friends could be main characters. They might come to check on Harry and his sister, bringing gifts and stories about their parents. These friends could offer guidance and protection when needed.
Ron and Hermione are likely to interact with them. Ron might be a bit more protective of the sister, while Hermione would probably study and share knowledge with her. They could all be in the Gryffindor common room, discussing spells and strategies.
Voldemort's remaining followers could be antagonists. They might target Harry's little sister thinking they can use her to get to Harry. They could try to kidnap her or use dark magic on her.
In a 'Harry Potter and His Little Sister' fanfiction, the sister could be a new student at Hogwarts when Harry is in his later years. She struggles to fit in because of being Harry Potter's sister. Harry has to help her deal with the pressure, make friends, and also face the threat of a new dark wizard who has a vendetta against the Potter family.
One possible plot could be that Lemon discovers she has a special magical ability that is different from Harry's. For example, she might be able to communicate with magical creatures in a unique way. Harry then has to protect her from those who might want to use her ability for bad purposes.
In the original series, Harry is an only child, so the very concept of a sister is a huge deviation. This changes the family dynamics completely. Instead of just having the Dursleys as his only family situation outside of Hogwarts, now there is this sister factor.
One possibility is that Harry discovers a long - lost sister and takes on the role of a protector. He might teach her magic in secret, away from the prying eyes of the wizarding world authorities. Maybe they live in a hidden cottage and he passes on the skills he learned at Hogwarts to her.