I wouldn't because I do not think it will break down.
You can put all lawn cuttings in with your compost don't put to much together try putting paper and cardboard in layers to add carbon to the mix.
Ok, thanks for your help......good job I asked!
Wrightt is right!! Moss won't rot down, it will just go dormant, and when it reappears from your compost heap it will come alive again. If it is going to end up at the bottom of a planting hole then it will do no harm and will help to retain moisture. If you use it as a mulch then you will quickly get a new patch of moss.
I use my lawn moss to line hanging baskets.They last for ages & I just top them up with fresh each year. I put them in a large bin bag and spread it out to the width of the bag, sprinkle of water and compress it before tying it up. After a couple of days, - or weeks,- I just put my fingers down the sides about 3-4" then slide them under to ease out a round pad of moss that almost fits the basket, (should get bigger bags really to get a better fit she thinks ) then fill in the gaps once a plastic liner, (recycled compost bags cut to shape) and a bit of compost to keep it in shape. If it ever stops raining i'll take a pic and post it here