The room is finished with drywall as is the garage but no air conditioning. There is a vent to the roof but no fan.

Is the "vent to the roof" simply the chimney for the water heater? Is the mechanical room vented otherwise? (Seems like code would require both a low and a high air vent for that closet separate from the exhaust stack)

there are three vents - one for water heater, one for radiant heat furnace and one for the room itself. I assume the radiant heat is causing the warmth in the pantry because it looks like the pipes go out in that direction. So, I guess I answered my down problem. We have a hot water recirculator with a timer so maybe we can cut back the recirculation times a bit more. Sure seems like a poor home design to me - one of many. The kitchen is on the west side of the house and the master bedroom windows are facing east. Should have realize all of this before purchase.

Is your radiant heat furnace turned on now?

Maybe try to better ventilate the room? I can't completely follow your description (a photo or diagram would help) but providing air flow to remove heat from any type of furnace or mechanical room takes two vents, an in (low) and an out (high).

Poor (or thoughtless) design or building elements that create issues for home owners is VERY common in houses, surprisingly so.

"The room is finished with drywall as is the garage but no air conditioning. There is a vent to the roof but no fan."

Is there insulation in the drywall between the mechanical room and the pantry? If not, that could be your starting point. Sounds like you have a lot of heat generated in the mechanical room. You might want to consider a radiant barrier also (between the insulation and the mechanical room drywall).

How do I determine if the wall is insulated without punching a hole in it?