Final answer:
Turning off the heat when leaving the house for several hours is more energy-efficient. Heat loss is slower when the temperature inside is closer to the outside temperature, leading to less energy use to reheat the house upon return.
Step-by-step explanation:
Strictly in terms of energy conservation, it is more efficient to turn your heat off when leaving the house for several hours in winter, especially if you can use a timer to make sure your house heats up by the time you return. This is because the heating system would not have to replace all the heat lost to the outside during your absence. Heat loss occurs faster when there is a greater temperature difference between the inside and outside of a house. Therefore, if you allow the inside temperature to drop (without reaching levels that could cause damage such as freezing pipes), you minimize the difference and thus reduce the rate of heat loss. When you return, the heating system will use energy to raise the temperature, but the total energy consumed will likely be less than the energy used to maintain a higher temperature all day, especially in less insulated homes.
Homes with better insulation or those designed as zero-energy houses will keep the heat longer and minimize energy use for heating or cooling. However, even in these homes, turning off or lowering the heat when not in residence can save energy. The principle is to match the power required to maintain a desirable temperature (or temperature difference, ΔT) with the time the heating system is running. In conclusion, your neighbor's suggestion aligns better with energy-saving strategies.