I know that comedy and drama are both very old genres in theater, dating back to ancient Greece at least, and it seems likely that film borrowed from those genres in theater. However, I'm not familiar with any theater performances that would fall under the horror genre. The only things I can think of that films might have borrowed from are books like Frankenstein. So were there precursors to horror on the stage or did film draw from other sources?
A grand old tradition. Almost as old as the English theater. Macbeth. A lot of the Jacobean Cannon.
The White Devil by Webster is the first play that I can think of that uses the old "you think the villan's dead, but then he pops back to life" trope.
I always thought that Shakespeare's Titus Andronicus was reminiscent of a poorly-plotted slasher flick.
That said, I think the modern horror film owes more of it's themes and tropes to literature than to theater. A lot of the themes that would be prominent in 20th century horror films can be seen reflected in a glass darkly, so to speak, in the Gothic horror fiction of writers like Edgar Allen Poe and Sheridan Le Fanu.
H.P. Lovecraft was also pretty influential on the creation of the modern Horror genre, writing a famous treatise on the subject entitled Supernatural Horror in Literature, which begins with the memorable opening line:
The oldest and strongest emotion of mankind is fear, and the oldest and strongest kind of fear is fear of the unknown.