He has lung cancer metastasis for bones. he is under chemotherapy and the doctor did a biopsy on a lymph node on the left side of his neck. now he is complaining of left arm numbness. what could be its reason?

Many things: Too numerous the possibilities. Most important is to discuss with his comprehensive cancer team. After any procedure, it is best to have open communication with your physician team. Cause of numbness or pain can be minor or serious. It would be a disservice to guess/speculate in such serious and complex disease. Our best wishes are with you during difficult time.

Likely nerve injury: Suggest that the nerve related issues may been either trapped by the nodes or may have been injured by neck positioning or biopsy needle itself.

Call the oncologist: Did he have brain or spine metastasis? Call the oncologist and inform them of the new symptoms, unlikely related to the lymph node biopsy.

Or something else: If neck and arm pain continues to worsen or develops into weakness or numbness and tingling you might consider MRI of the head and c-spine to make sure that there isn't also a separate lesion that could be causing this. While a deep neck biopsy possibly could cause some nerve damage, most simple fine needle aspirated of a palpable cervical neck node shouldn't cause such. Could consider new pet?