Is it permissible for a woman to take birth control pills without her husband's consent, if she is harmed by frequent childbirth and the husband's neglect and unwillingness to discuss the matter, and would her action be considered a sin?

Procreation is a shared right between spouses. It is permissible to delay it by mutual agreement for a valid reason, but neither spouse may arbitrarily prevent it. If the wife is severely harmed by frequent pregnancies to an unbearable extent, and the husband cannot convince her to use contraception, she may use contraception as needed, based on the Prophet's (peace be upon him) saying: "There should be no harm nor reciprocating harm."