Snape would likely be angry at first. Harry has always been a source of trouble in his eyes, and running away would just add to that perception. He might mutter some sarcastic remarks under his breath about Potter's irresponsibility.
Snape might suspect that there is some dark magic at play. Given his knowledge and expertise in the Dark Arts, he would immediately look into any signs of foul play. He could also be conflicted because on one hand, Harry is a student he has had many issues with, but on the other hand, he is also an important figure in the fight against Voldemort. So he would have to decide whether to report Harry's disappearance or try to handle it privately.
In this kind of fanfiction scenario, Snape might initially be conflicted. On one hand, he has his long - standing dislike for James Potter, Harry's father. But on the other, his sense of duty as a Hogwarts professor could kick in. He might reluctantly start to heal Harry, all the while muttering under his breath about how much trouble Harry always seems to get into, just like his father. Then, as he heals Harry, he might start to see some of Lily, Harry's mother, in him and soften a bit.
He could run to the Forbidden Forest. It's a place full of mystery and danger, but also a good hiding place as it's vast and not many people would expect him to be there. He might find some solace among the magical creatures there.
One place to start looking for the missing fanfiction could be on large fanfiction platforms like FanFiction.net or Archive of Our Own. They have extensive collections of Harry Potter fanfictions. Check their search functions, maybe filter by characters (Snape and Harry Potter) and relevant tags.
If Snape finds Harry Potter fanfiction, he'd probably be quite irritated. He has a rather stern and no - nonsense personality. He might see it as a frivolous waste of time and be disdainful of the creative liberties often taken in fanfiction. He'd likely scoff at the idea of people writing about his students and the wizarding world in such unregulated ways.
Look for any unique plot details you remember. If it involves Snape finding Harry Potter in a particular place, like a hidden chamber or a different magical realm, use that as a search keyword. For example, if it was in a 'forbidden forest' setting, search for 'Snape finds Harry Potter in the Forbidden Forest fanfiction'.
Maybe he wants to escape the constant danger in the UK wizarding world. Voldemort's followers might still be a threat even after his defeat, and France could seem like a safe haven.
One possible reason could be that he feels unsafe at Hogwarts. Maybe there are threats from dark wizards within the school that the teachers are unable to fully protect him from. Sirius, being his godfather, represents a safe haven.