No, taking birth control pills does not cause any chronic issues with fertility. In fact, studies have shown that women who have recently ceased birth control have the same chances of getting pregnant as women in the general population. You can take birth control for years and still be just as fertile as you were beforehand. That being said, it may take you a few weeks or months before your body returns to its naturally fertile state.
The reason it takes some time for women to become fertile again after birth control is that your body is adjusting to the lack of hormonal supplementation. Since many birth controls work by stopping ovulation, changing the cervical mucus, and thinning out your uterine lining so an egg can’t implant, you need all three of these factors to return to normal before you can get pregnant. As your body begins to rebalance its hormones, the return of these three processes will restore your fertility.
However, for many women, fertility returns immediately after ceasing birth control, so don’t have unprotected sex right away unless you’re ready to get pregnant.