Is not taking my mother to perform Umrah—due to her health conditions and need for assistance, and my desire not to bear additional responsibility—considered disobedience and failure to honor her?

You are not obligated to accede to your mother's request to take her for 'Umrah if your health conditions and hers do not permit it. Obedience to parents is obligatory in matters that are beneficial to them and do not cause harm to the child. This should be done while managing her expectations, placating her, and seeking help from close relatives to convince her.