I’m no air quality or air movement expert, but I would have to expect that the open bay door exchanges air with the outside at a far greater rate than the filter processes its air. So, while the effect can’t be zero, I would think that running it with the door open isn’t accomplishing much.
The good news is that the air from the outside is “fresh” air, so you may be doing more for your air quality by opening the door than you are by running the filter. Even better, point a fan through the doorway.