How do I remove my ex-husband's name from my children's birth certificate if he is not the biological father?

Your ex is technically the legal father of your twin daughters and is obligated to support them at this point in time.  If, however, you do not believe that he is the biological father you have to bring an action seeking to establish paternity.  Once the paternity test is done and it has been established that your ex is not the father the Court will order that his name be removed from the birth certificate.  All legal obligations to support the children will cease at that point as will all rights to inheritance, etc.