Is it obligatory for a sister to divorce her Shiite husband, and what is the ruling on her remaining with him, and what is the fate of the children, and is there sin in that, given their knowledge of Shiite beliefs?

Affliction is a divine law in life, by which Allah raises ranks and expiates sins. It is the path of prophets and the righteous. Through patience and seeking reward in the face of affliction, a Muslim attains the reward of the patient. As for one who cannot practice their religion, they must emigrate. If that is not possible, then they are considered oppressed, and it is obligatory for them to manifest what they can of their religion.

Regarding your sister's stay with her Rafidhi (Shiite) husband: if he adheres to beliefs of shirk (polytheism) and kufr (disbelief), then it is not permissible for her to remain with him, based on Allah's saying: (Then if you ascertain that they are believers, do not return them to the disbelievers; they are not lawful [as wives] for them, nor are they lawful [as husbands] for them). If she is able to invite him to the Sunnah and succeeds, and he remains with her, then the marriage contract must be renewed. However, if she is unable to invite him and fears for herself, she should seek separation and deliverance, while trying to take her children. If she cannot separate out of fear for herself, then she is considered among the oppressed, and she must try to distance herself from intimacy with him as much as possible.