I would say it depends on the type of costume, the weather and the child. My DD8 was very gun-ho about wearing her Belle costume for the entirety of the 24 Hour Rock Your Disney Side event back in May. After our Fast Pass to meet Mickey (our 1st time!), the temperature felt like it shot up by 20 degrees! Then after we got through Main Street it went up another 10 and the poor thing looked so sad when she looked at her mom and I and asked "can I take it off".
Of course we weren't going to force her to wear it, we had brought a change of clothes and put on something much cooler.
If it was a lighter costume she might have toughed it out, but princess costumes very breathable, and the heat was unbearable that day.