I have a couple of the splatter screens that you can buy to cover a pan to keep the grease from splattering out of your pans... I put it on top of a box or something where the air can circulate around it and put my pieces on it. They dry through and get hard. My daughter has a lot of things that I have made for our grandson's cakes and she has them sitting on his dresser and some are 2 1/2 years old and still look great! There is some stuff that you can put on them to preserve them too. I have a bottle of it but have never used it. It has a very strong smell.
First make sure that you remove any butter cream from the piece. Then you can place it on a covered cake board. You can either keep it in a cake box in a cool dry place or you can display it under a glass/plastic dome/box. Just remember that the color will fade over time if displayed in the light. You can slow this process by keeping it away from sunlight or bright lights.