Have the owner run the heat for you. Do not turn on a Heat pump in the heat mode if it's above 75°F degrees outside. It may damage the unit. As with the other systems of the house, if there is no power or fuel, be suspicious. Look for soot, heat damage, rust, or physical damage to the equipment. Are there dangling or damaged controls or obvious amateur repairs? Is the system old or antiquated? With all air systems check for damaged ducts and missing filters. Oil fired systems usually have an underground tank that may leak with age.
Are the bricks in the fireplace loose or the mortar damaged? Soot up the face of the fireplace may indicate that it does not draw well. Look for cracks in the brickwork that might allow fire to escape. If there's a wood stove or insert you won't be able to see inside the fireplace or flue. Have it checked before closing. Look for weather damage to the chimney.