During the middle of the summer, when the heat really builds, and rain becomes scarce, we really see that the lawns take a beating. They dry out, turn brown and go dormant. However, that shade on a summer day is not only nice for you, it provides plenty of cooling for the grass.
Water is lost through the leaf blades through transpiration. During days that are hotter, the amount of water lost is greater. However, when a tree provides shade, it cools off the grass, thus slowing the transpiration rate. This helps to keep the grass greener.
Another way to fight transpiration is through irrigation. Since rainfall can become lacking, or hard to predict when it will happen next, irrigation can help to supplement the water lost throughout the day. But there are times when the lawns would need vast amounts of water just to stay green. This is when we can really see the benefits of shade for grass.
This may be the one benefit that a tree can provide for grass, otherwise they are in constant competition. Usually the trees and grass are competing for sunlight, moisture, nutrients, space, etc.
Just like we enjoy the shade on a warm summer day, so does the grass. The grass that does not receive any shade will definitely need some help through the warm days of summer.