Is it possible to run an Illustrator (or Photoshop) action without running the processes for displaying it? I assume this would speed up the performance—or maybe this is minimal.
by "running the processes for displaying", I am assuming you mean you can see each step of the action being run. Well, photoshop or illustrator actions are just a shortcut way of doing repetitive things without you having to do them manually.
If you have an action to create square and center it. The software does each step on its own on your artboard. It selects the tool to create a square and creates the square. There is no background process where the action calculates the end result and renders the output on your artboard. Each step is carried out in front of you. You can also pause the action while it does so. You could pause the action before the software centers the square.
Neither app has a "headless" version. They both rely on the GUI.
However, you might want to choose the Playback Options in the Action Panel Menu and ensure the Accelerated option is ticked to make things run faster. (Note Playback Options is available in both apps.. same location)
That way you don't have to sit through every step... it plays back at a faster rate when possible.
You can run illustrator and photoshop hidden via the COM interface (Visual basic) on Windows. There is however no speed benefit that i have noticed, inDesign is in fact a bit faster. Mostly because it has a better designed and more modern api.
Scripts are faster than actions anyway, as they do not echo individual steps like actions do. But if your intention is to run actions then calling actions on hidden window is just as slow as it is without the update. There is a intentional slowdown in actions that is independent of your refresh rate.