Caring for the large exposed roots that break through the surface of the soil will protect the health of the entire tree.
The large, woody roots that commonly emerge through the surface of the soil primarily function as anchors for the tree. These roots have feeder roots growing off of them. Both types of roots take in oxygen for the tree. Trees often send roots above the surface for greater access to oxygen. Covering these roots incorrectly will only starve the tree and lead to issues with rot. Cutting the roots to remove them can cause part of the trunk or crown to die.
It's not always best to leave large exposed roots completely uncovered either. Hitting a root with your lawn mower can leave a large wound, which invites disease into the heart of the tree and attracts pests. Using a light layer of mulch or well-draining soil over the roots will cover them just enough to protect them from damage by helping mowers and other equipment to roll over the roots without clipping them.
Adding a layer of coarse, organic mulch around a mature tree can also can slow down water evaporation, keeping the tree healthy during the driest days of summer.
If you do decide to cover exposed roots with soil or mulch, watch for the development of mushrooms. If the roots are smothered and die due to overly thick cover, fungus can rot the roots and enter the rest of the tree. Keeping the surface roots and base of the tree dry and exposed can keep this fungus under control.