Well, it might develop with Harry having strange dreams. Since he has a connection to Voldemort, his sleep state could be a gateway to some sort of shared mental space or prophetic dreams. And Snape would then be left to interpret these dreams when Harry wakes up.
Another way it could develop is that Harry's friends notice his absence and start to search for him. This could lead to a series of events where they suspect Snape has done something wrong, but as the story unfolds, they realize there was a good reason behind it. Maybe Hermione uses her research skills to figure out what's really going on.
The story could also take a more magical turn. With Harry asleep, the magic around him might start to act in strange ways. Snape could be trying to control these magical fluxes, and in the process, he discovers new aspects of magic. This could then lead to a new adventure or quest where they need to use this newfound knowledge to defeat a new threat in the wizarding world.
It could also be for some sort of magical experiment or study. Snape, being a Potions master and a very knowledgeable wizard, might need Harry in a dormant state to study something about his unique magical abilities. For example, his connection to Voldemort might behave differently when he is asleep, and Snape wants to observe this.
At first, it's likely to be tense. Harry has years of seeing Snape as a bully. But as they start living together, they begin to understand each other. Snape might show his more vulnerable side to Harry, like his memories of Harry's mother. Harry, in turn, starts to respect Snape's knowledge and sacrifice. Eventually, they develop a bond like a father - son relationship.
Often, it develops into a more complex and strained relationship. Harry might start to resent Snape even more, while Snape might feel guilty deep down. They could end up in a cycle of mutual dislike and misunderstanding.
In fanfiction, their relationship can develop in many ways. Sometimes it starts with Harry being suspicious of Snape as in the original books, but then Snape shows his selfless side. For example, he might save Harry from a dark spell and that starts to change Harry's view.
At first, it's likely to be very tense. Snape has always had a grudge against Harry because of his father. But as they live together, they start to see each other's true selves. Harry might see Snape's vulnerability and intelligence, while Snape might see Harry's courage and determination that are different from James's arrogance.
Often, it starts with intense dislike. Snape is constantly critical of Harry in class, deducting points for the slightest mistakes. Harry, in turn, sees Snape as a bully. As the story progresses, there might be small moments of grudging respect, like when Harry realizes Snape's intelligence in potions, but the overall hatred remains for a long time.
At first, it could be very tense. Harry might be reluctant to submit, but as time goes on, he starts to see Snape in a different light. This could lead to a grudging respect. For example, Snape might show Harry some advanced potions techniques that no one else has taught him, which makes Harry realize Snape's value.
Their relationship could develop from pure hatred to grudging respect. Harry may come to understand Snape's difficult position in the wizarding world and Snape might see Harry as more than just James Potter's son. For example, Snape could start to appreciate Harry's courage and determination during their time together in the summer.
Their relationship development is complex. Snape, being Harry's guardian, has to deal with his own past issues regarding Harry's parents. Initially, Harry is very suspicious of Snape's motives. However, as time goes on, Snape starts to share some of his knowledge with Harry, especially about potions and the darker aspects of magic that he thinks Harry needs to know for self - defense. This sharing of knowledge gradually breaks down the barriers between them. Also, Snape might start to involve Harry in some of his personal projects related to protecting Hogwarts, which makes Harry understand Snape's dedication to the school and to him in a new light.
Their relationship develops in a rather interesting way. Snape, being a strict mentor, forces Harry and Draco to look past their differences. Harry begins to understand Snape's past and the reasons for his actions, which in turn makes him more open to Draco. Draco, who has always been in Snape's good graces to some extent, has to adjust to sharing Snape's attention with Harry. This causes some initial jealousy but later leads to a sort of rivalry - turned - friendship as they both grow under Snape's tutelage.