CCUK
Free Flying
Chickens will eventually finish off your lawn but the Moss will be caused by shaded damp areas. I had moss all over my front lawn and could get rid of it so o had it paved. Problem solved! No grass for chickens though.NO. What you are describing is not the fault of your chickens. It is the fault of poor soil management, and perhaps not enough sunlight. Are the trees that you mention shading your lawn?
Chickens will scratch up the sod. They will make dust baths here and there. They will not cause grass to be replaced by moss.
If you have a lot of shade, that may need to be fixed. I suggest that you get a soil test done. Moss is favored by a low pH. Your lawn fix may be as simple as keeping the birds off it for a while. During this time, you would adjust pH with proper application of lime, fertilizer, and over seeding with a grass mix appropriate to your micro-climate.