Can wisdom teeth pain affect face structure?

Pain: Generally no unless there is significant swelling of the area however this is temporary until the issue is solved by the dentist. If by structure you mean a permanent change no.

It won't unless: The infection pierces through the cheek like any other infected teeth esp. Bicuspid or molar.

No, but........: Pain from problematic wisdom teeth cannot permanently affect facial structures, other than the temporary facial grimaces that you might make. However, wisdom teeth can become infected which in turn can become acute and cause facial swelling and bone destruction. Once resolved, your body generally heals this. I had a patient who developed a cyst in the lower jaw that left him permanently numb.

NO: No. Just temporary swelling. Back to exactly the way you were after healing. See an oral surgeon for expert wisdom tooth advice.

It is possible: Cyst around the wisdom tooth can damage the structure of jawbone. For example keratocystic odontogenic tumor.