This is not really an answer, since I don't think this can be answered without being on site. Rather, I'm going to explain a bit about how the system works, and offer a few ideas as to what the problem could be.
To start, you'll want to understand how air moves through the system. You'll want to know the difference between conditioned air, and combustion air. And you'll want to understand the normal ignition sequence of the unit.
Combustion air, is the air the furnace uses to burn fuel. As you may know, fire requires heat, fuel, and oxygen (oxidizing agent).
Heat is typically supplied using a hot surface igniter, spark gap, and/or a pilot light.
In this case, fuel is natural gas.
Oxygen comes by way of combustion air. In this case, the inducer draws air into the combustion chamber, and expels it out through the exhaust vent.
Conditioned air, is the air within the building. The furnace draws cold air into the system through return ducts. The air is heated in the furnace, and then returned to the living space through the supply ducting.
NOTE: The furnace you have is installed in what's known as the downdraft orientation. This means that the return air enters at the top, and moves down through the furnace.
The above image illustrates how the different types of air moves through the system.
Rather than type the whole thing out, I've copied the sequence directly from the installation manual. I've also highlighted the section dealing with the pressure switch.
Now that you; hopefully, have a better understanding of how the system works, I'm going to cover some of the things that could be going wrong.
Lack of combustion air
After seeing the installation, my main concern is that the system isn't able to draw enough combustion air. Typically when furnaces are installed in utility closets, there are combustion air intakes that allow air to enter from non-habitable space. If the furnace is starving for air, the pressure in the system will likely not reach the threshold to close the pressure switch.
You mentioned in a comment that the exhaust is shared with a water heater. I'm not sure how likely it is, but it's possible if the vent is not sized properly, that this could cause a problem. When the water heater is on (heating), it could be creating a situation where the furnace can't achieve the proper pressure to close the pressure switch.
Blocked and/or restricted vent
If the exhaust vent is blocked or restricted in any way, that could cause a problem. This could include junk stuck in the pipe, broken/damaged pipe, too long of a pipe, too many bends in the pipe, etc.
If you want to try and tackle the problem yourself, I'd probably start by leaving the closet doors open (clearing the stuff from inside the closet, might also be a good idea). The next step would be to inspect the exhaust pipe, but it sounds like you don't want to tackle that yourself. So if running the unit with the closet doors open doesn't help, it might be time to contact a local HVAC technician.
I apologize for the long post.