There are almost never seedlings growing from acorns. If there were, you would be able to pull them up easily. What you see are sprouts from the roots of the existing tree; therefore, you do not want to spray a herbicide on them in an attempt to eliminate the sprouts for you will hurt the “mother” tree along with the sprouts.
Only a small percentage of oaks send up suckers from the roots. It is a genetic trait, like freckles, except I like freckles. But like freckles and sunshine, some trees have the ability to sucker, but do not unless stimulated to do so. Oaks having a slight tendency to sprout suckers will often do so when roots hit a barrier, such as trees confined to a parking lot planter, or between a sidewalk and driveway. Also, when roots are disturbed and damaged by rototilling, they are more likely to sprout suckers. But some trees never will make suckers. When choosing an oak in a garden center, if there are sprouts coming up at the inside edges of the container, I would avoid that tree.
You may choose to mow them along with the grass, if grass still exists. Or if the grass has thinned too much, you might plant Asiatic Jasmine groundcover, and use hedge trimmers to trim the jasmine and oak sprouts to a uniform height. You can cover the area of sprouts with a heavy gauge woven geotextile, and then either mulch or spread large gravel or decomposed granite over the top of the geotextile. My favorite solution, when appropriate, is to cover the ground with geotextile and then build a wood deck.
Or if you prefer a thick green lawn, you may remove the oak tree, and all of the tree roots with a backhoe. If you just cut down the tree, grind down the stump and all the large roots you can see, there will still be thousands of oak sprouts emerging from the remaining roots in your new lawn or bed area for a few years afterwards. The area will need to be continually sprayed with an herbicide.
James 
marilyn ann guidry says
10 years ago a landscaper applied geotextile material and a large amount of rocks underneath my large 26 year old oak tree. Thousands of little oak trees appeared around this area and a very large area around the tree and killed my grass. I was told the rocks were killing my tree for lack of air and water. I paid a guy to take the material and rocks out. There is a large pill of death roots or old grown oak seedling present now. Should they all be cut out ? Where do I go from here. The landscaper told me he could use a dig deep and remove the seedlings and put down some more St Augustine grass after trimming the dead limbs off this tree and the other for more light. I live in Austin and since the cold spell we had in Jan they have gotten very bad. I keep them trimmed but the area looks black and very unattractive. Thanks for your assistance.
james says
This sounds like a pretty complex situation that I can’t understand or make recommendations on remotely. I would search for good reviews on a certified arborist and get them to take a look.
Haywood Jablome says
The sprouts were so bad around my oak they towered over the jasmine, broke through mortar and dislodged stones in the planter ring wall. It required regular trimming akin to a hedge. I tried burning and excavating to no avail. It sprouts through thick landscape fabric and grows in mounds. After the hard winter, I cut them to the ground, burned the edges and dug out the mounds with a mattock. Then I covered the area in cheap plywood, stapled landscape fabric over that and covered it under a thick mulch bed. It still bursts out of the seams around the planter and at the trunk.
I have another oak 50 feet away similarly planted with a nice bed of jasmine and no sprouts. This tree must be a genetic anomaly.