Thank you. After wisdom tooth removal, a sharp pain at the side of tongue while swallowing is usually from a small ulcer or the tongue rubbing over a sharp bony edge/socket margin left after extraction. Sometimes it is minor irritation of the nerve near that area. It settles in most cases but the dentist has to feel that area once.
Next Steps
Warm salt water gargle 3 times a day, soft cool diet, avoid spicy food. A local anaesthetic gel (like Mucopain) before meals helps. Show the dentist who did the extraction if pain is same after 4-5 days, or earlier if you get fever, swelling of jaw or trouble opening mouth.