No, it doesn't need to be removed and yes, it can be composted. Many folks intentionally plant moss in this area in shade gardens, since it does so well and grows so prolifically :-) Shade loving perennials and bulbs will easily grow through it.

More than likely what comes off that tree is Lichens and not moss, although both are first stage plants Ma Nature grows to add organic matter to soil so higher order plants can grow in that soil. Lichens grow on trees where dust from the air accumulates in the bark and traps just enough moisture for those plants to grow. What is knocked off by the birds can be left where it falls, mixed into the soil, or picked up and composted.