What are the dangers of birth control pills in an attempt to regulate heavy periods in a 40 year old woman?

Very few: Unless there is a history of blood clotting, liver cancer, stroke, or cardiac events the use of episodic birth control pills are usually safe for temporary control of heavy cycles as long as the cause has been determined by the doctor.

Varies: There is always a small risk of blood clots while taking the pill, fortunately very small. The risks otherwise depend on your general health. High blood pressure, migraines, diabetes, gallbladder disease could be aggravated. You should not take it if you smoke.