Moss lawns tend to be safe for dogs – particularly varieties such as Irish moss – as they are non-toxic. Thanks to their unique rooting system, they can handle a fair amount of wear and tear (usually more so than grass), and a mossy lawn can provide a soft, cool surface for your dog to lie down on, too.
All that being said, moss lawns can be damaged by heavy footfall and pet waste, so it's worth weighing up the pros and cons – because dogs using the lawn as a toilet is one of the main reasons for having to deal with a yellow lawn.