Just purchased a home. Inspector said I should somehow ventilate the furnace room. The furnace room is very small and the water heater is also in it. Its about 4' X 4'. Any cheap ways to ventilate it that will not be dangerous? The doors are hollow.

Usually a louver in the door is the solution, but that depends what's on the other side of the door.

The Uniform Mechanical Code states that in unusually tight construction, vented appliances can be installed in a mechanical room if the room contains at least 50 cubic feet of volume for each 1,000 BTUs of fuel input. A combustion air opening can be sized at 1 sq. inch per 3,000 BTUs of fuel input. This opening must be in the upper 12 inches of the room and needs to freely communicate with the outdoors. Direct-vent appliances are vented directly to the outside, usually through a wall, and they get combustion air the same way.

In ordinary construction where gravity warm-air or fan-assist furnaces will be installed in a mechanical room, two required openings of 1 sq. inch per 1,000 BTUs of fuel input are required, one located 12 inches off the floor and the other within 12 inches of the ceiling. Gravity warm-air or fan-assisted furnaces are those appliances that vent out through the roof—normally through a chimney—and do not bring in combustion air from the outside.

A cheap way to ventilate which is not dangerous is to cut a hole in the walls (drywall) or hollow core door. Install a grill over the holes for cosmetic reasons. See comment above for size and placement of holes (within 12 inches of ceiling and one within 12 inches of floor). Usually about 16x16 is good.

Your question made it sound to me like you might think cutting the hole would be dangerous? Look out for wiring and plumbing inside the wall, otherwise, not cutting the hole is dangerous. The purpose is to provide plenty of fresh air to the gas furnace and water heater for proper combustion and draft up the chimney. As a temporary fix, leave the door open for now.

The other side of the door is the Hallway to our bedrooms and bathroom, about 4 foot wide and 10 feet long. I was worried with the hollow doors that if I made a hole in it, and I just put a grill over the front for cosmetics, if the warm air got inside the hollow door, I could cause a fire. Any thoughts?

My guess is the surface of the door is just as combustible as the inside of the door. Furance sides probably require a couple of inches clearance to any combustible (wood, paper, drywall). I think cutting a hole will be fine for safety. It may be a little noisy by the bedrooms, but creating carbon monoxide due to lack of fresh air is worse than noise.