Is there any problem with using continuous birth control, that makes it that a woman never gets her period?
There certainly is an issue with it. The reason is because the first mitzva written in the Torah is Pru Urvu to be fruitful and multiply, therefore any form of birth control is causing a violation of this commandment. A person who abstains from fulfilling this commandment is considered akin to murdering, as the result is the same- it is causing a loss of life. Even after the couple have fulfilled this commandment and bore a son and daughter, there is a rabbinic obligation based on the verse “l’sheves yitzara”- that the world was created to be inhabited, that the couple should still have more children. Granted that when necessary, taking pills is a preferred form of birth control; because it does not involve causing wasted seed, nevertheless it is still not correct unless there is a specific need. When there is a need, it should be discussed with one’s local Rov.
Shulchan Aruch Even Haezer 1-1.