Lovecraft was significantly influenced by Edgar Allen Poe and you can certainly see that influence in his gothic horror stories. Contemporary writers like Stephen King, Neil Gaiman, William S. Burroughs, and Alan Moore have all cited Lovecraft as having a major influence on their writings. Argentine writer Jorge Luis Borges wrote his short story There Are More Things in memory of Lovecraft. Even writer Joyce Carol Oates is a fan - she wrote an introduction for a collection of Lovecraft stories.
In the realm of film, John Carpenter and Guillermo del Toro have used Lovecraftian images and themes in their films. Surreal visual artist H. R. Geiger, the man who designed the aliens in the James Cameron Alien films, cites Lovecraft as an inspiration for many of his designs.
While many of us know Arkham from the Batman canon, Arkham is actually a fictional place in Lovecraft's stories. The necronomicon, the book of the dead, is mentioned in a number of sci-fi stories, most notably in Sam Raimi's Evil Dead movies. It's also a Lovecraft creation.
Mr. Lovecraft was known to encourage the use of his fictional creatures and places in other works, so I think he'd approve of Cameron McNary's play.