Am I religiously obligated to serve my husband's family members who reside with me for more than three nights, especially given that they do not assist me with housework and I am suffering from back pain?

Hospitality is among the noblest of manners and a sign of faith. The majority of scholars have held that it is a Sunnah (recommended) and not obligatory, while some have maintained its obligation for a day and a night.

As for a wife serving her husband's family, it is not religiously obligatory upon her. However, she ought to do so out of respect for her husband. And the husband should not compel his wife to do what is not religiously obligatory upon her, and he should help and assist her in that.