No. Cast iron maker Lodge says its bare cast iron should be washed by hand, because a dishwasher will remove the seasoning and likely cause rust.
Seasoning is the thin layer of baked-on oil that gives cast iron its dark, slick, rust-resistant surface. A dishwasher cycle combines a long soak, hot water, and harsh detergent, and the pan then sits wet while it cools, which strips that layer and leaves bare iron exposed to moisture. Lodge lists dishwashers alongside soaking in the sink, air drying, and damp storage as the usual causes of rust. Its recommended routine has three steps: wash with warm, soapy water and a stiff brush or scrubbing pad, dry the pan completely with a towel, and wipe on a thin layer of oil. Mild dish soap is fine, but soaking is not. If a pan does go through the dishwasher and rusts, it can be saved: scour off the rust with steel wool, wash and dry it, oil it inside and out, and for heavier rust bake it upside down at 450 to 500°F for one hour to reseason it.
Source: Lodge Cast Iron, Cleaning your Lodge cookware
Image: Griswold cast iron skillet, by NMorales (WMF), CC BY-SA 4.0, via Wikimedia Commons.