Can a rotator cuff tear cause arm and hand numbness?

Not usually. The shoulder is usually an 'innocent bystander' when it comes to neurologic problems such as numbness and weakness that is found beyond the shoulder in the arm and forearm. The more likely culprit for these complaints is actually nerve compression emanating from the cervical spine.
Possible/unlikely. If there is a significant amount of swelling around the nerves that pass through to the arm and hand, a rotator cuff tear could cause those symptoms. It is far more likely that a nerve is being pinched- either in the neck or tight muscles around the neck/shoulder. A physiatrist can diagnose and treat these conditions.

Related Questions

Cause of unilateral edema from possible rotator cuff tear.; edema to neck, arm to hand and LLE 2+pitting. CRP always elevated, + ANA normalforedema?

Clot considered. Even though blood clots in the arm are not very common, swelling of the arm, especially from the upper arm to the hand raises the suspicion for a blood clot. Lymphedema is possible as well. Rotator cuff problems typically do not cause widespread swelling. The edema is not often visible to the naked eye and remains around the tendon. It's time to see a doc very soon, preferably in the next 24 hours. Read more...

Rotator cuff pain, arm and hand goes numb is that tendon damage?

Anterior subluxation. Of the shoulder can be accompanied by rotator cuff pathology but pain and discomfort in the the shoulder with certain positons that is accompanied by tiingling and or numbness the hand transiently is often a sign of anterior loaxity in the shoulder, the so called apprehension sign. Read more...
Pinched Nerve. Sounds like you have a pinched nerve in your neck. I would suggest seeing a pain/spine specialist to evaluate you further to evaluate what level and what can be done to treat you. There are interventional treatments besides medications and surgeries that might reduce or eliminate the pain altogether. Read more...

Rotator cuff tear-pain in elbow-wrist-hand- swollen forearm-catch in bicep, bicep knotted up, when pushed, it has a cracking popping feeling, painful?

Multiple Sources. I suspect you have multifactorial pain meaning you likely have shoulder pathology involving the biceps tendon and rotator cuff but I am also concerned you have other sources for pain given your distal hand/forearm swelling. There is an entity called hand/shoulder syndrome that includes a frozen shoulder...See your doctor. http://www.theshouldercenter.com/shoulder-specialist.htm. Read more...