Is it against the law for a non-custodial parent to not know where their child lives?

Depending on the specific circumstances of your case, for example the terms of the custody agreement, this could be deemed to be a case of parental kidnapping/interference.  This constitutes both federal and state offenses and can be punishable by hefty fines and/or incarceration.

Additionally, if the custodial parent denies the other parent visitation, custody be changed.  Denial of visitation alone does not support a custody change. However, visitation interference, coupled with other acts such as a decision to change the child's surname or (as in your case) not informing the non-custodial parent of the child's whereabouts, which are found to be outward manifestations of a comprehensive scheme by the mother to deny the child any type of relationship whatsoever with his father, may constitute a sufficient change of circumstances affecting the child's well being.

Right now you need to consult with an attorney in your area; as well as report this to the police.