What would happen if someone covered a small hole in the fuselage of a plane with a soft body part?
I assume the hole has been properly deburred, otherwise you could suffer cuts on the sharp edges of the hole.

What happens else depends on altitude and flight speed.

At low level, the outside temperature should be tolerable and the pressure difference between inside and outside should be small, so your hand will be fine.

At higher altitude, outside air temperature can drop to -60°C rsp. -76°F. Since air density is lower, the aircraft will travel fast (approx. 250 m/s rsp. 560 Mph in case of an airliner), and convective cooling at the hole will soon cause hypothermia and frost bites. Also, inside pressure is 0.6 bar rsp 8.8 lb/sq-in higher than outside pressure, and the exposed skin on your palm will feel this as a sucking sensation. 

I further assume that you need to remove all cabin insulation to put your hand over that hole, so your whole hand will be exposed to the airplane's metal skin which will have the same temperature as outside air. If the hand is dry and does not freeze to the metal skin, you might manage to keep it in place for a few seconds, but it will hurt. Keep it in contact with the metal skin longer and you will suffer progressive frostbite. The part of the skin which covers the hole should suffer less, since convective cooling should be less intense than the thermal conductivity of the metal skin, which will quickly pull all body heat from your hand where it touches the metal.

The pressure difference will not damage your hand, but the low temperature will.