Interview with the Vampire could fit into a few genres, if you want to get really specific. Examples: Homoerotic Vampire Romance. Broody Vampire Whine-fest. We could go on.
Unlike Anne Rice, we're just going to go vanilla and call this baby a horror novel and hope it plays well with others. Horror novels don't need to be gory, though Interview does offer plenty of blood and death to fill that niche.
What horror novels need to do is evoke a feeling of dread in both the characters and the reader, and "dread" is the perfect word to describe the anticipation of what's going to happen to Louis, to Claudia, and even to Lestat. There's a ton of foreshadowing in this book, and foreshadowing is a key component in any good horror novel.
A large subgenre of horror is called Southern Gothic. Think Faulkner's "A Rose for Emily" if little ol' Em were an immortal vampire. Interview is set in New Orleans for most of the book, but being set in Nawlins alone doesn't make it Southern Gothic. However, Anne Rice's depictions of the slaves, the slaves' attitude toward black magic, the presence of vampires, and the idea that you have no idea what these creepy white folk are doing in their huge kudzu-covered plantations make Interview the perfect candidate for this genre as well.