A door and the metal doorknob on it are obviously at the same room temperature. Yet the doorknob feels colder to the touch than the door. Why?

Metal conducts heat at least 500 times better than wood.
Heat travels from a hotter body to a cooler body.
The better the conductor the quicker the transfer
Also the greater the temperature difference between the hot and cold bodies the quicker the heat transfer.

Compared to the door or handle (at room temperature) your hand is the warmer body. The heat transfer from your hand to the metal handle is relatively fast but to the wood of the door it is much slower. So the handle feels colder than the door.
If the door and handle were at 100 degrees centigrade and you touched them then the door would feel cooler than the handle.
They would both actually be hot, but the speed at which the heat is transferred into your fingers would be much slower from the wood and you probably would not get burned. The metal handle might give you blisters!

See any of many explantions of heat conduction.