Ask Mr. Smarty Plants

Yes. Absolutely. Don't do that.

Sod is greatly overdone, especially under oaks. Oaks, including live oaks, are allelopathic, which means they exude substances to inhibit growth of other plants competing for light, water and air beneath them. Along with that, there is the shade that such a large tree casts over the grass. So, the grass is doomed in two ways.

If that ground is tilled up, it will damage and expose to air roots 2 to 3 times farther out than the crown of the tree. Exposing and damaging those roots makes them very vulnerable to Oak Wilt. The process of working around the tree can also cause damage to the bark. When an oak tree takes damage of that sort, it begins to leak sap. Oak Sap is milk and honey to the Nitidulid Beetle which does not, itself, harm the tree. However, in the land of Oak Wilt, there are other damaged oaks around oozing sap; those trees are very likely already infected with oak wilt. So, the beetle will have a sap snack, get the Oak Wilt fungus all over him, and then move on to your damaged and oozing tree and, as he sips sap, he will be infecting your oak with the same fungus. And the tree is doomed.

Our suggestion is a nice spread of a good-quality shredded bark mulch, over as much of the circumference of the oak roots as you can manage. It will help the roots stay cool or warm in the  different seasons, hold in moisture and, as it decomposes, blend in with the soil to improve its quality and provide nutrients. It is attractive, smells good and helps inhibit weeds.

If the oak has to be pruned for any reason, and you can't wait until December and January, when the nitidulid beetle is having his winter siesta, or you must prune because of wind damage etc, be sure that every cut bigger around than your thumb is coating with pruning paint.

Please read the articles on Oak Wilt from Texas Oak Wilt.com in  which the Lady Bird Johnson Wildflower Center is an active partner. Your oak is probably Quercus fusiformis (Escarpment live oak), follow the link to learn more about it.