Birch, 28, is best known for her work in the films "Ghost World" and "American Beauty."
The Post said Wednesday she lost her most recent job because the cast and crew of "Dracula" didn't get along with her manager/father, former porn star Jack Birch, who insiders say was constantly hanging around during rehearsals.
The elder Birch allegedly threatened an unidentified co-star during a scene because he was rubbing his daughter's back, the newspaper said.
"(Director Paul Alexander) didn't want to do this. Everybody loved Thora and thought she was great," a show insider told the Post. "But (Jack Birch) threatened a cast member and (Alexander) didn't want something like that, that would wreck the whole show."
Emily Bridges, the daughter of veteran actor Beau Bridges and Birch's understudy, will take over the role of Lucy from Birch in the play when it opens Jan. 5 at Manhattan's Little Shubert Theatre, the newspaper said.