There's various ways to incorporate modern technology into horror, depending on what subgenre of horror you're aiming for.
With telephones, sure, you can have a mobile/cell phone run of battery or lose signal at an inopportune moment, or a storm knock out the power for landlines, but active use of the phone can facilitate horror. The phone ringing at wrong moment has already been mentioned. Having the villain call a potential victim to terrorise them first is another option, as demonstrated by the opening of every Scream movie, but the second. Needing to ditch a phone because it can be tracked is a staple of modern espionage fiction, but that principle can be employed here too. More subtle uses are also possible, like the phone ringing simply when it's not expected (eg. late at night), or shouldn't even be possible (eg. turned off, or line severed), such as in The Black Phone.
Audio-visual recording equipment is great for building tension and fear. From Ringu (The Ring) to Ghostbusters 2 to The Brown Lady of Raynham Hall, cameras capturing something supernatural has long been established. Maybe the camera picks up something that isn't visible to the naked eye, or the opposite: something you see and even interact with doesn't get recorded. Was it a glitch or something supernatural? Are you just losing your mind, or is there a scientific explanation? Same can go for audio recordings.
Remember those stupid chain letters that said you had to send it to 10 other people or Something Bad™ would happen? Remember how that transitioned to email in the early days of commercial internet? Well, what if it wasn't naughty word?
Now imagine the premise of Ringu combined with YouTube and chain emails. That's basically a horror pandemic.
Buffy the Vampire Slayer was mentioned in the OP and there were several episodes that utilised modern technology to propel the plot. For example, in the episode I Robot, You Jane, the demon Moloch had been trapped in a tome back in the 15th century, but in the modern day, Willow was trying to digitise Giles' collection (which just so happened to include this book) for more efficient indexing and in scanning this particular tome, Moloch was released into the internet. He was able to get people to build him a robotic body to transfer to (hence the episode's title), but that's rather less interesting than a demon having access to near enough every computer system - a nice twist on the rogue AI escaping into the internet.
