The short answer is yes. Though there are examples of clay pieces lasting hundreds of years, small parts like ears, legs, etc. can be fragile. When attached to a wood base or other material, cracks can appear in the clay piece due to differing expansion temperatures, so they shouldn't be exposed to extreme heat or cold. All fine art should be kept in temperature controlled environments, but that’s especially important for a clay piece that’s attached to a base. See my How To Sculpt Clay post for more information on the clay medium.