Q: I know that a theology degree can prevent you from being admitted into some countries. Would a "closed country" overlook this and admit you because they want English Teachers?

It depends on how bad that country wants teachers. I would say yes for some countries. Our country requires a copy of the undergrad certificate. If the name of your school isn't openly Christian the theology degree can be explained as a study of a philosophy of living. That has been a successful and well-understood explanation for some of our co-workers.