I recall from when I was into making figurines with Sculpey that the recommendation was to heat it in a well-ventilated place because of the fumes. If they could hurt people, I would assume they would be extra harmful to birds.
However, Scupley is a polymer clay, I think. Not a natural clay. Not sure about playdough. It is pretty benign. You can even eat it. I'm sure the fumes will be different depending on the type of clay and some would be safer than others. But I would still recommend baking the clay in a well-ventilated space away from your birds.
When I did clay art, I bought a cheap toaster oven from Goodwill and set it up outside on my porch when I was ready to cook some clay. It kept it out of my kitchen and away from any appliances used for food preparation. Perhaps you have somewhere like that?