Yes, they eat it, they trample it, they poop on it (which can actually burn plants when fresh - but is great for plants once composted). There's no such thing as a beautiful green lawn inside a chicken run! If you do want to keep grass under them you would need to put them in a movable tractor and move it every day or every other day.
The really nice thing about it though is that they are natural mulchers. We put leaves and yard clippings in their enclosure on a regular basis. They shred them by scratching around and then we throw the results onto the compost heap. Speeds up the composting process tremendously.