I think it depends on the yarn and how colorfast their dye is. some bleeds a little, some a lot. If you do it in the washer I'd use an old pair of jeans destined for the rag bin with it. that way if the color bleeds and stains, it wont matter.
You will probably then want to run a cycle of the machine again with just the jeans or old towels to sort of clean out the machine.
I like to put the thing to be felted in a zippered pillowcase, to reduce the woolie bits getting stuck in the machine. but it can be a pain in the *ss to take it out, unzip and look at it every 5 minutes.
did I say one project at a time?