Leather climbing shoes
Washing climbing shoes in a washing machine: Not recommended
Exception: Made specifically to be washed, is the La Sportiva Oxygym:
If you are positive your shoes are only made of synthetic material (and not leather), you can risk it. Tips: use cold/warm water, low spin, mild detergent (aka no bleach) put the shoes in a pillowcase to avoid the rubber hitting the walls of the machine and make sure the laces/velcro is undone or very loose so the water can get in the shoe.
Your risks are: shoes stretching, stitching/glue coming loose due to heavy spin or heat.
Washing climbing shoes by hand: Recommended
You can wash your shoes in the sink, using a mild detergent*, and may use your hands or a brush to assist in rinsing the shoes. Stuff your shoes with newspaper as they dry.
*If the shoe is made of synthetics, you can usually get away with a more intense hand-washing soap and/or heavier rubbing.