Informal definitions here: In any system, results (output) which influence further actions (input) constitute a "Closed Loop". If the results of actions do not directly influence further input, you have an "Open Loop".
Strictly speaking, an open loop is no loop at all. In a closed loop system, the output "loops back" and in some way effects the input.
Common examples: Cruise control on a vehicle is a closed loop - changes to load, etc, are 'fed back' into the control system to effect how much the vehicle must accelerate or decelerate to maintain a relatively constant speed.
If you pay no attention to what effect your marketing has on your sales, and make no changes to it based on that, you have an "Open Loop".
These phrases have been around at least since the dawn of computer science. I don't know "Getting things done", but in most cases a closed loop would certainly be part of effective management.