Putting dawn dish soap in your dishwasher is not recommended however it won't do much harm to use dawn dish soap in your dishwasher.
The main difference between dishwasher detergent and regular soap such as dawn dish soap is that the dishwasher detergent creates less suds in the dishwasher than regular dish soap does.
So when you put the dawn dish soap in the dishwasher it can create more suds in the dishwasher which can take a bit more time to rinse the soap off the dishes.
I have used dawn dish soap before in the dishwasher and it did okay but I had to run the dishwasher through a second rinse cycle to clear off all the soap.
But other than that it didn't do any harm to the dishwasher.