Polyester shirts can shrink in the dryer if you use the heat setting on the dryer.
If you want to dry your polyester shirts or other polyester in the dryer you should use the cool setting on the dryer and not the heat setting.
I've always used the cool setting on my dryer when drying polyester.
Polyester Garments can go in the dryer.
When drying polyester in the dryer set the dryer to the cool setting so that the polyester does not shrink or melt.
Polyester is not regular fabric and not cotton so you need to dry the polyester in the dryer on the cool setting or it could also melt.
Polyester is a man made fabric that is made of a plastic and when being dried in the heat setting of the dryer the polyester could melt.
Shrinking polyester is hard to do under normal circumstances but sometimes the polyester may shrink in the dryer especially if you use the heat setting.
Or you could just hang the Polyester out to dry outside on a clothesline or in the house.
But using the cool setting on your clothes dryer you can safely dry your polyester.