It will work with the door open but conditions won't be ideal. At the very least, try and find a way to get airflow out the back, either by moving the desk away from the wall or by cutting at least a 4-inch hole toward the rear of the compartment, near the PC's exhaust fan(s). Expect increased dust buildup and less-effective cooling compared to other configurations, both because of the proximity to the carpet as well as the enclosed nature of the setup.

You could modify the enclosure to optimize cooling if you're up for a fairly basic DIY project. It would involve fully closing off the back, then adding a filtered intake in the door and an exhaust duct in the rear.