Can a torn rotator cuff cause hand numbness when sleeping?

No: These are 2 different diagnoses. Tingling in the hand at night comes a nerve impingement. A torn rotator cuff can cause shouldered pain and limited mobility of the shoulder.

Yes, although rare: Numbness after a torn rotator cuff would most likely result from a change in sleeping posture. Often after the injury, patients pre- and postoperatively tend to sleep with the injured shoulder up when they sleep on their side. Sleeping on one's side, often with hands tucked under the pillow or wrists flexed is a very common cause or exacerbating factor in median neuritis or carpal tunnel syndrome.