One possible reason could be Snape's unfair treatment of Harry and his friends. Snape often showed blatant favoritism towards Slytherin students and was overly harsh on Gryffindors, especially Harry. This constant bias could lead Harry to want him fired. Another reason might be Snape's connection to the Dark Arts. Harry, having faced Voldemort, might be worried about Snape's knowledge and use of dark magic within Hogwarts. Also, Snape's generally unpleasant demeanor towards students could be a factor. His sarcastic remarks and cold attitude might make Harry think that he is not a suitable teacher for Hogwarts students.
Well, it could be simply because of the bad blood between them. Snape has been tormenting Harry since his first day at Hogwarts. He constantly deducts points from Gryffindor for the smallest of reasons when it comes to Harry and his friends. This continuous harassment could make Harry believe that Snape is not a good influence in the school. Also, Harry might think that Snape is not teaching Potions effectively. We know that Snape is brilliant at Potions, but his teaching style might not be conducive to the students' learning. He may be too strict and not allow enough creativity or exploration in the subject, which could lead Harry to want him out of the teaching position.
Maybe Harry wants Snape fired because of Snape's past associations. Harry discovers that Snape was once a Death Eater. Even though Dumbledore vouches for Snape's change of heart, Harry might still be suspicious. He could fear that Snape might relapse or that his past could endanger the students. In addition, Snape's role in the Triwizard Tournament mystery, where he seemed to be involved in some shady business, could also fuel Harry's desire to have him removed from Hogwarts. Harry is always looking out for the safety of his friends and the school as a whole, and Snape's actions make him seem like a liability.