Can a high dental filling cause swelling of the gums & tooth sensitivity? if so & not fixed promptly what problems can arise? thanks

Yes: But so can extrusion of the opposing tooth, gum disease, dental abscess, root fracture, and other possibilities. Please see your Dentist for appropriate examination, imaging, and tests in order to determine cause of your problem and appropriate treatment.

High dental filling: High dental filling needs to be corrected ASAP. Initially tooth may become hyperemic and sensitive to hot and cold or sensitive as you chew. It may be mistaken that you need a root canal treatment but simple bite adjustment may be a culprit. Not corrected will result in “overworked tooth” bone loss around the tooth, or variety of things like fracture of opposing tooth or the tooth itself.