IF there are not two pipes coming out of the top of the furnace (intake and exhaust), it is relying on drawing combustion air from the room. Having it in a closet without at least the minimum air opening is a real problem whether it is the only problem, it's hard to say.
When you looked, did you see any error indication on the furnace? FWIW, if the WH turned on during that time, that additional burner may have still exceeded the available air needed, and caused the furnace to lock out.
The following is from a State WH installation instructions manual, but is pretty common for all burners that do not have closed combustion (ie.,getting their combustion air from directly outside).
The confined space shall be provided with two permanent openings communicating directly with an additional room(s) of sufficient volume so that the combined volume of all spaces meets the criteria for an unconfined space. The total input of all gas utilization equipment installed in the combined space shall be considered in making this determination. Each opening shall have a minimum free area of one square inch per 1,000 Btu per hour of the total input rating of all gas utilization equipment in the confined space, but not less than 100 square inches. One opening shall commence within 12 inches of the top and one commencing within 12 inches of the bottom of the enclosure.
Note, if you use louvers...they would need to be much larger than that minimum of 100sqin because the louvers block a lot of that area...you need free air passage. That passageway to unconfined space still h asvolume requirements, and if that is too small, you might need to duct it to outside. That requirement is at least 50CUFT/1K BTU...so, if your WH has a 40K BTU burner, you'd need those openings into an area that had at least 2000CUFT of volume - with an 8' ceiling, that's a room a little bigger than 12'x12'. If your furnace is 120K and the WH is 40K, you would then have 160K worth of burner, and need 160sqinx2 of opening in the door, and it must connect to a free space of 50*160=8,000CUFT, or at an 8' ceiling, 100'x100'! I'm certain your house isn't that large! This means, drawing it from outside would be the only option.
FWIW, part of getting an inspection is to have followed both the local and national codes, but those dictate the appliance must be installed per the manufacturer's instructions.
As we tighten up our houses, open combustion devices make it harder and harder, which is why things are moving to closed combustion burners. Throw in a big gas stove, maybe a gas insert fireplace, then turn on your range hood and bathroom fans...you can be sucking all sorts of outside air in through any cracks...and, as the house gets tighter, that gets harder and harder. Older houses weren't that much of a problem, because they often leaked like a sieve, at least in air. This is why new homes built in more enlightened places (Canada comes to mind), require fresh air ventilation that would help ensure both a means to provide fresh air to vent out excess moisture but also to bring in enough air so your combustion devices can not create a lower pressure in the house.