Is it common to permanently lose all sensation in your chin after having jaw surgery?

Not common: What kind of jaw surgery? Bilateral surgery? Very occasionally a difficult impaction of a lower wisdom tooth with full grown roots that are on, very near to or wrapped around the mandibular nerve, can cause loss of sensation for while (rarely forever), usually referred to as parathesia. Usually it is one side of the jaw in this situation. If you had a chin procedure, it could be bilateral.

See below: Talk to your jaw surgeon. The nerve could have been damaged during the surgery, or by the initial trauma. It should slowly resolve.

Jaw Surgery: After orthognathic jaw surgery, mainly a bilateral sagittal split osteotomy, it is not uncommon to have a sensory disturbance in your chin. It may stem from normal postoperative swelling or it may be iatrogenic in nature. Sensation returns over time in a majority of the cases. I encourage you to discuss with your oral and maxillofacial surgeon.