Is there a reason that dryers aren't grounded
The metal body of the appliance is connected to ground.

The problem is that the clothes are charging you with respect to ground as you unload them. When you touch something grounded, then a large current flows to ground through your finger. You have a relatively large capacitance to ground, many 10s of pF, which stores quite enough energy at a few kV or 10s of kV for the discharge to be felt through your fingertips.

There are several things you can do about this.

As this is an electronics engineering site, the first thing to mention is what airline companies do to prevent the same problem, which is a discharge between the airplane and the fuel bowser, two large conductive objects on insulating tyres. You obviously do not want a spark between the fuel nozzle and the plane fuel filler! Before refuelling is started, a grounding lead is attached between them, to equalise their potentials.

Use an ESD strap, grounded to the drier, when you unload your clothes. Although each item of clothing will carry a small charge, the energy that's released by each small zap of the clothes to you will be too small to be felt. As the clothes are not conductive, you only get the charge from the tiny area in contact with your hand. The problem with the drier shock you describe is that you are conductive, and the shock you feel is the entire charge that was transferred to you, going all at once.

ESD straps are useful as well when you're working with static sensitive components. Remember you can kill a component with voltages and energies far too low to be noticed, even by your fingertips.

Another method is to anticipate that you will have been charged by the clothes, and discharge yourself by the knuckles to the drier before touching the door with your fingertips. This takes some anticipation, but you will learn fast. Knuckles have an order of magnitude or two less sensitivity than fingertips to an electric shock