Ask Mr. Smarty Plants

First we want to refer you to an all-purpose previous Mr. Smarty Plants answer that has links in it to a lot of other answers, all on the subject of oak sprouts and what to do in order to try to control them.

Please don't indiscriminately spray herbicide, any kind. It will drift onto other plants, not only your plants, but your neighbors. It will contaminate your soil and go into the drainage into the water supply. It is pollution on a large scale, and will not do your particular problem any good. The shoots that you have coming up are the attempts by that tree to survive. Each shoot that comes up quickly develops some leaves, which then begin to manufacture nutrition for the tree. As far as those roots are concerned, that tree is still alive. Spraying herbicide might brown some of the shoots (and kill a whole lot of other stuff) but the root is protected by the earth. We would recommend contacting a licensed arborist and having the roots ground out, since you are going to have to level to pour the concrete or whatever you are using for the driveway anyhow.

Even grinding out the roots will not totally eliminate the sprouts or suckers. There will still be enough viable root to get those sprouts going. Frequently the appearance of suckers occurs when the tree sustains some damage, like being cut down. But when you pour the concrete, that will stop the shoots from coming up, right there, anyway. Stay vigilant at pulling out, clipping off or mowing the shoots. Eventually, the roots still alive in the ground will starve to death because by destroying the shoots you have denied those roots the food they need to survive.