This is the greatest thing that any horror story can achieve, it is also one of the most mind-numbingly difficult things to do. Of the horror movies which have achieved this coveted state, I will state only a handful have truly done so. For whatever reason, possibly because people need to use their imagination more, I'd argue horror literature has an easier time of achieving immersion. I'll still list my top three immersion examples from movies, though, because they're references people can immediately get an idea of what I mean from.
Jaws, of all things, is the probably greatest movie success for building immersion. No, people didn't believe the events depicted on-screen actually happened but they believed Great White Sharks were capable of killing people. The tourism industry as a whole suffered that summer and the author of the book devoted rest of his life to preserving sharks due to the hysteria he inadvertently helped create.
Prior to Jaws was Orson Welles 1938 radio drama broadcast of the H.G. Wells classic, The War of the Worlds. Even before the internet was invented, Orson Welles managed to successfully troll a substantial portion of America. However many were actually fooled is anyone's guess, probably not as many as would be cinematic, but enough that it obviously had an affect.
Finally, more recently, there was The Blair Witch Project where a lot of gullible Americans were fooled by the idea of found footage. Less gullible Americans still appreciated the attention to detail that went into making the story plausible if not believed.
Most modern-day horror films, I tend to think aren't actually horror per say. They're action or fantasy features which just happen to have a higher gore count than what is expected by audiences. Routinely, after seeing a bunch of his buddies slaughtered, the hero rallies back and defeats the monster in the end...