I take levora (ethinyl estradiol and levonorgestrel) birth control pills, can I start taking pure garcinia cambogia extract weight loss pills or will it make levora (ethinyl estradiol and levonorgestrel) stop working properly?

Who knows? Most of these supplements are not well regulated so you never know exactly what you are taking. Most have not been well studied so how they will effect other medications can be uncertain. If you want to lose weight do what we know works : decrease calories and increase exercise.