Plastic Dishes can generally be washed in the dishwasher safely as long as the plastic dishes say they're dishwasher safe.
When washing plastic in the dishwasher be sure to put the plastic on the top rack away from the heating element because if you happen to place the plastic dishes on the bottom rack the plastic dishes could move around and even land near the heating element and melt.
The heat being so close to the bottom rack also can melt some plastic dishes but when you place the plastic dishes on the top rack in the dishwasher they should wash just fine and not melt from the heating element.
I've washed several plastic dishes in my dishwasher without problems but I did once put a plastic cup on the bottom rack accidentally and it melted because the heating element that dries the dishes got the plastic cup too hot.