Do PDO threads have a risk of nerve damage near mouth/jaw?

Thanks for questions.
PDO thread is placed subcutaneous layer and nerve is located deeper than fatty layer. Technically it is possible  but it is very rare to have nerve damage. Occasionally local anesthesia given during procedure can cause temporary weakness but it usually recover once the local effect fades away.

Dr.Choi

The efficacy and safety of any procedure is related to the experience of the surgeon performing the procedure.  If you go to a provider who has minimal knowledge of the anatomy of the facial skin and soft tissue layers then you have a higher incidence of injury to the nerves or vessels of the face.  In the hands of an experienced provider a thread lift should have minimal potential for any adverse neurological side effects as the layer the threads are placed are very superficial to the plane at which the nerves and major vessels exist.  

Threads are inserted along the subcutaneous plane just below the skin. The important nerves that you refer to, are much deeper below the fatty layer in this region. Therefore nerve damage is extremely rare especially when done by a trained physician.