Given Snape's own experiences of being bullied and misunderstood, he might be both repulsed and fascinated by the Dursleys. He could end up teaching them a lesson in humility through magic, like making Dudley experience what it's like to be small and powerless for a while, just to show them how Harry feels all the time.
Snape would likely be disdainful towards the Dursleys. He has little patience for Muggles in general, and the Dursleys' mean - spiritedness would not endear them to him. He might use his cold, sarcastic tone on them, leaving them confused and a bit frightened.
The Dursleys would probably be more frightened. They are muggles and Snape is a powerful wizard. The idea of a strange man in black robes suddenly appearing and using magic would be terrifying for them.
Well, Daphne in the 'Daphne meets the Dursleys fanfiction' would probably interact with the Dursleys in a very cautious way at the start. She would be observing their behavior towards Harry closely. Once she realizes the extent of their nastiness, she could try different approaches. She might try to reason with them, explaining how wrong it is to treat a child like that. When that doesn't work, she could use her magical abilities to disrupt their daily routine in a harmless but annoying way, like making the TV constantly change channels or the kettle never boil. This would be her way of getting back at them for their treatment of Harry and also a way to show them that there are forces they don't understand.
He might be very angry with them at first for how they've treated Harry.
Remus might interact with the Dursleys in a very calm and collected manner. He would probably be polite, despite their rudeness.
Perhaps Snape would initially mistake the Dursleys for some sort of Muggles with a bit of dark magic in them due to their cruel nature. He could start investigating them, which would cause chaos in the Dursley household as they try to hide their secrets from a wizard like Snape.
In fanfiction, if Snape stays at the Dursleys, it could have a huge impact on Harry. Snape could start to teach Harry some basic magic skills early on. This would not only give Harry an advantage when he finally goes to Hogwarts but also change the way he views Snape. Harry might see Snape as a mentor figure rather than just a strict teacher. Additionally, Snape could share stories about Harry's mother, Lily, which would strengthen Harry's connection to his family heritage. Snape's presence could also prevent the Dursleys from starving or mistreating Harry as much as they do in the original story, leading to a healthier and more confident Harry.
If Snape stays at the Dursleys in fanfiction, it could lead to some interesting character interactions. Snape might have a complex relationship with Dudley. He could try to teach Dudley a lesson about bullying, using his Slytherin - like cunning. Or perhaps he'd end up having some sort of begrudging respect for Vernon Dursley's determination to keep the wizarding world out of his family's life, while at the same time finding it completely absurd. And with Petunia, there could be a whole range of emotions, from her fear of him as a wizard to a possible shared understanding of their complex feelings towards Lily Potter.
Perhaps Snape uses magic to create a distraction at the Dursleys' place. He could make strange noises or apparitions that scare the Dursleys, and while they are in a panic, he grabs Harry and disapparates. This would be a quick and efficient way for him to rescue Harry.
The Marauders would likely have a very negative reaction to the Dursleys. Sirius, having run away from his own awful family, would not tolerate the Dursleys' treatment of Harry. He'd probably confront them right away.
In the fanfic, Snape might use his knowledge of the Dursleys' house layout. He may enter the house unnoticed, find Harry, and then use a Portkey to quickly get them both out of there. He could also have some sort of charm or potion that he uses to counter any magic the Dursleys might have used to keep Harry there, like a spell - breaking potion. And he may have planned the whole rescue very carefully, choosing a time when the Dursleys are least likely to notice, like in the middle of the night.