Absolutely not, birth control pill, patches, ring, implant, IUD with hormone, do NOT cause infertility in the future. I think the confusion arises in that 1 out of 10 women will have PCOS. PCOS causes irregular periods and those women will have difficulty getting pregnant later. The treatment for PCOS is birth control pills. So we put these women on birth control pills and when they come off, they don’t remember they had PCOS or they weren’t diagnosed with it BEFORE they were put on the birth control pills, so when then try to get pregnant, they blame the pill. When really it was the PCOS that caused the difficulty getting pregnant.
Also, ironically enough, women with PCOS are MOST fertile the moment they get off the pill because that is when their hormones are most normal. Their hormones get more abnormal (more PCOS-y) the more days they are off the pill.
Read more about PCOS here https://www.pandiahealth.com/polycystic-ovary-syndrome-what-is-it-and-how-do-you-treat-pcos/