I refrigerate mine all the time. They will sweat a little, but give them an hour to acclimate and they’ll dry right out. If there’s hand-painting or something that CAN’T run, cover the cake with a plastic bag ( tape the opening of the bag to the cake board ) and it won’t really sweat at all.
Jenniffer White, Cup a Dee Cakes - http://cupadeecakes.blogspot.com