Certified GKH Gardening Expert
I would actually suspect that a lack of water is the issue. Many parts of the country have experienced very hot and dry weather. Many plants that did not previously need watering during the summer now need to have supplemental water. With trees, when they become water stressed, they will literally suck every drop of water out of the ground around them, which makes it twice as difficult for plants to survive under them.
Try watering the tree. Make sure it is a good deep watering each time you do it. The tree has likely used up even the deepest reserves of water in the soil and it is killing both the tree and the grass.