Edited to Update:
I washed Nip the Cat with a load of clothes. Regular detergent. He came out great. His whiskers (that were yarn-like) are a little frazzled, but he came out really nice. I didn't throw him in the dryer because he was pretty dry when I got him out of the washer. (And I figured that the beans would melt!) His tail was a little damp, but I am sure that it will air dry just fine.
So I would say that yes, you can wash Beanie Babies in the washing machine! :)
I am not sure, but I am going to do a load of clothes today and I will throw one in...I will let you know! :)