I have a live oak tree that is about 12 years old (20 feet tall). The ground around it has sunk a couple of inches to a diameter of about 10 feet. Is this cause for concern? And, will it be harmful to the tree roots if I add soil to the top to bring the level of the lawn back up? I live in Dallas, where we have lots of clay and are in a drought.

I would not touch it. If you add dirt you may be harming the tree's ability to exchange air and water. A tree too high is better than too low.
Tony M