Will taking the birth control pill cause higher amounts of estrogen in the body?

Yes, if you are taking a combination estrogen/progesterone pill. If progesterone only then the answer would be No

Oral bcp's contain both estrogen and progesterone derivatives, but today's bcp's have very low amounts of both, and taking them will stop the production of ovarian estrogen/progesterone. The level of estrogen in your body is lower on bcp's than that which would be produced in the ovary (w/o pills) after about age 25, and both hormones are cleared during the 7-day placebo period.