Is it permissible for the husband to remain in his marriage after his family demands that he divorce his wife, and is the wife considered sinful if the relationship continues?

There is no sin upon you in the continuation of the marriage, as you have committed no wrong towards your husband or his family. Obedience to parents is only obligatory in what is rightful and good, and their command to divorce a wife without cause is not from what is rightful and good. Therefore, your husband is not obligated to obey them in divorcing you, and there is no sin upon you for remaining his wife. He is obligated to honor his parents, and it would be good for you to help him in that.