On the other hand, Harry could be proud of himself for standing up for what he believes in. He may think that he has made Hogwarts a better place by getting rid of a teacher who was not treating students fairly. But at the same time, he might have mixed emotions as Snape was also an important part of the Hogwarts environment for so long.
Hermione could be a great help. She is very intelligent and resourceful. She could help Harry find evidence against Snape. For instance, if Snape is suspected of cheating in grading, Hermione could use her knowledge of Hogwarts' grading system and records to dig up proof.
In this kind of fanfiction, perhaps Harry discovers some serious misdeeds by Snape that lead to his dismissal. For example, Snape might be caught unfairly favoring Slytherin students in a major way, like giving them unearned high marks in exams while unduly punishing Gryffindors. Harry, being the observant and just character he is, would report this to the proper authorities at Hogwarts, which could be Dumbledore or the school board. And this could start the process of Snape getting fired.
Well, he could use the power of friendship. Maybe with the help of his friends like Hermione and Ron, they come up with a plan. They might find evidence that Snape has been involved in some really bad things that go against the wizarding laws. Once they present this evidence to the proper authorities, like Dumbledore or the Ministry of Magic, Snape could be gotten rid of. Also, Harry might have some sort of prophetic dream that gives him the key to getting rid of Snape, just like he had dreams about Voldemort in the original series.
In some fanfictions, it could be that the author has rewritten Snape's character in a more approachable way. Maybe Snape shows a side of himself that is protective towards Harry, like secretly guiding him in his magic studies without the usual harshness, which makes Harry feel safe.
In some fanfictions, Snape might use his advanced knowledge of potions to create a special concoction to heal Harry. He could draw on his skills in Herbology as well to find rare ingredients for the potion.
For Harry Potter, it could be a mix of emotions. On one hand, he might be relieved as Snape has been tough on him. But on the other hand, he might also feel a bit guilty or confused, especially if he later discovers that Snape was wrongly fired. Hermione would probably be outraged at the injustice of it all and might try to find evidence to get Snape reinstated.
One possibility is that the fanfic changes Snape's backstory to make him more sympathetic. Harry, being empathetic, then feels safe around him. For instance, if Snape's past is shown to be full of hardships that made him act the way he did, Harry might understand and feel safer in his presence.
Perhaps Harry witnesses Snape being extremely unfair to a group of students, like constantly giving them detention for no good reason. Harry decides to take a stand and gathers statements from the students. He then goes to the Hogwarts board of governors and presents his case, leading to Snape's dismissal.
It can lead to a completely different dynamic. Harry might have a new perspective on Snape and the magical world.
Dumbledore would probably first have a long conversation with Harry. He would want to understand Harry's concerns thoroughly. Then, he would talk to Snape privately. Dumbledore is very protective of Hogwarts and its students. If he found that Snape's actions were truly endangering the students or the school's harmony, he might consider some form of disciplinary action, but firing Snape might not be his first option. He would also weigh Snape's contributions to the Order of the Phoenix and the fight against Voldemort. He has a lot of faith in Snape, but he also respects Harry's instincts and would take his feelings into account.