As a hirer, I would never do this. If I needed to call some references, I would ask the candidate to provide a list.
As someone who goes over a lot of resumes, if I know a director on someone's resume, I will contact them. I do this ALL the time.
Here's the problem with references, these are the three people picked by the person applying for the job. I bet you right now, those three people are going to say good things about them. Why bother calling them? I will usually only contact them if I am trying to see if they fit the job, or if I want to get some more information and I think a third party is better then the person actually applying (this is far more for educational positions, such as interns.) But after calling a bunch of references last year, I actually stopped calling them - unless I personally know the person or I am on the fence about the applicant in someone.
If a theatre is listed, I would contact their production manager. If a director is listed, I would feel free to contact them. If the list a SM they worked under, I would contact them. UNLESS there was some notion on the cover letter or resume that they were doing a confidential search - and didn't want to notify them - at which point I may ask the person if I can contact further people.
The lesson here is if you had a bad show, a bad relationship with a director or theatre, you might want to leave that show off your resume. I think in this business it happens all the time people will contact people they know off their resumes. Heck, I also just call up a stage manager I know who might know them to get a vibe on the person.
In the end, it's all about the personal connection. I could have two very talented stage managers applying for the same position, but one person I may get along with, and one I might not - if I know the person who I am getting a reference from, then they will know me and hopefully give me a honest, but more educated reference. Someone telling me "She's great, but you two would not get along" is a much better reference then "She is great."