Solution found! It turns out that on my garage door opener, the wire had come loose from the back of the electronic eye/sensor beam that shoots across the garage about 6" off the ground.
The purpose of this device is to make sure something (like a child or pet) is not in the way of the door when it is closing. As the wire was loose, the circuit was broken, and the opener was always thinking that something was in the way. As soon as I fixed the loose connection, the problem went away.