Is it a requirement to have a solid core door opening into a furnace closet?

I'm not aware of a part of the Residential code that would specify door materials. Design of the door can make a difference though. Louvered doors or doors with grill panels are often used for increasing the amount of combustion, ventilation and dilution air available to the furnace. An existing door opening into the mechanical room is perhaps the easiest way to deliver more air if your calculated volume of the air in the room is insufficient for the size of the furnace. Louvered door makes it possible to have both: a door and an additional air inlet. See section G2407 of the International Residential Code.