In some fanfics, Snape might punish Harry because of his perceived insolence in class. Harry has a rebellious streak, and Snape, being strict, wouldn't tolerate any behavior that he deems disrespectful to his authority as a teacher.
Another reason might be that Harry is suspected of sneaking around in areas where he's not supposed to be. Snape, as a vigilant teacher, may punish Harry to maintain discipline in the school. He may think Harry is up to something related to the ongoing dangerous situations in the wizarding world and wants to keep him in line.
In some fanfictions, Snape might punish Harry by giving him extra detention tasks like scrubbing the cauldrons in the Potions classroom until they shine. This is a common form of punishment as it's within Snape's domain as the Potions master.
One possible reason could be that Snape fails to deliver crucial information. Voldemort is a very demanding and unforgiving master. If Snape doesn't give him accurate details about the Order of the Phoenix's plans, for example, Voldemort would surely be enraged and punish him.
One reason could be Harry's resemblance to his father. Snape had a long - standing grudge against James Potter, and seeing Harry might trigger those old feelings. In slash fanfiction, this can be exaggerated, with Snape being harsher on Harry because of his connection to James.
At first, Harry is likely to be angry. He might feel that Snape is being unfair and targeting him. Harry has a strong sense of justice, and when he believes he's being wrongly punished, he'll be resentful.
It could be that in the context of the fanfiction, Snape believes Harry has been disrespectful towards him or other teachers. For example, if Harry were to talk back or show disobedience in class, Snape would likely punish him as a way to assert his authority.
However, as the story progresses, Harry may start to understand the reasons behind Snape's actions. He could become more resigned and accept the punishment as a part of his growth. He might use the experience to become more cautious in his actions and more respectful of the rules.
One reason could be to save him. Maybe Harry is injured during a battle or some sort of magical accident, and Snape, despite their differences, steps in to carry him to safety. For example, if Harry is hit by a dark curse and Snape, with his knowledge of dark magic, knows he needs to get Harry to the infirmary quickly.
Harry could try to talk to Dumbledore about it. Dumbledore has a lot of influence and might be able to make Snape see the error of his ways.
One possible reason could be that Harry breaks school rules in the fanfiction. For example, if he ventures into the Forbidden Forest without permission. Dumbledore is strict about the safety of students and such disobedience might lead to punishment.