I wouldn’t. Dishwashers use a lot of two things: hot water and agitation. This combination can easily ruin vacuum seals in insulated lunch boxes and heat plastics to the point of leaching. I don’t think I would hesitate to throw my single-wall, steel Stanley Classic in the dishwasher, but it’s just as easy to clean it with a sponge and some dish soap.
As a general rule, spot-clean your lunch box with warm water and soap.