Should you leave the heat on in an empty house during the winter?

This depends on the quality of the home's insulation, the method used for heating (oil, gas, electric), the length of time them home is expected to be empty, the outside temperature, and many other variables, including the current price of the energy used.

Generally, in a home where a person leaves for work and returns daily (the usual situation for most heating systems), it is best to leave the heat at a single setting. Alternately, if one's home is equipped with a programmable thermostat (currently federal law for new installations), it can be beneficial to program a small dip in temperature during specific periods of time when the house will be unoccupied. The thinking behind this is that it is more efficient to maintain the house at a single temperature, than to repeatedly re-heat the entire house. The lower the outside temperature, the more this is the case. In sub-zero temperatures, it would take an incredible amount of energy to allow the indoor temperature to freefall during the day, and then reheat the whole house every night. This would result not only in higher heating costs, but increased wear and tear on the heating hardware itself. In warmer temperatures, this is less so.

If leaving for an extended absence, such as in the case of a vacation longer than a few days, or seasonal absence, then the heat should definitely be left off. However, be aware of freezing-related damage that may occur within the home. Without any internal heating, you might encounter problems that would normally not be an issue, such as Plumbing damage from water freezing in the pipes, etc. The longer the absence, the more energy and money will be saved by turning off the heat.

Air conditioning is another story. High humidity (which is also corrected by A/C systems) can damage paint, drywall, wallpaper, electronics, etc.