This depends on the material from which they are made. Typically you can recycle cases, and plastic packaging associated with your contacts but these should be handled differently than your run-of-the-mill plastic bottles and cans. Bausch & Lomb and Terracycle and have teamed up to develop a program that's specific for contact lens recycling. You can visit their resources to find drop-off locations, which are often found in optometrists offices.
Prior to the point of recycling your contact lens case it is of great importance that it be kept clean. This is an often-overlooked source of eye infections, the dirty lens case. Always remove all the contact lens solution when the case is not in use, clean it, and leave it upside down on a clean surface to dry. The case should be discarded after 3 months. No one wins an award for extra long usage of a lens case. Bio-film, an almost invisible film of debris and bacteria can form in contact lens cases after extended periods of use. Try changing out the lens cases after 3-4 months.