Great question, and there are a lot of different opinions.
One of the basic laws of physics will tell us that, yes, turning off your heating or hot water system during the night will reduce your energy losses.
- Heat loss is directly related to the difference in temperature between inside and outside your home: an exterior wall with 21C (70 degrees Fahrenheit) inside and 11C (52F) outside will lose ten times as much heat as a wall with 21C inside and 20C (68F) outside.
- Lower your indoor temperature and you will lose less heat to the outside; common sense.
Where it gets sticky is in your perception of heat or warmth, and your perception is what's important.
- Imagine a room with 21C (70F) indoor air temperature, and it's been at that temperature for hours. All interior furniture and walls are at a similar temperature to the air.
- Now imagine that room after a cold night with no heat. You've turned on the heat and after a while, the air has warmed up to 21C (70F) but the interior surfaces lag behind and are still cold.
- Standing next to a cold surface makes you feel colder (try your bare arm next to a cold window).
So what do people often do? Crank up the heat a bit more until the interior surfaces warm up, and now you've lost much of what you've saved by turning down the heat overnight.
This is where programmable smart thermostats really shine. You can program your thermostat to turn off the heat when you head for bed and turn on the heat early enough in the morning that your home feels warm when you get up.
- You won't save quite as much energy as leaving the heat off til morning but you can still save quite a bit.
- There are also timers available for electric water heaters though they require professional installation because of the high voltage and current draw.
I've read different researchers claims of anywhere from 3% to 20% savings on your energy bills just from installing a programmable thermostat, making it possibly the simplest, most economical single thing you can do to save energy.
For more information:
Read "How much energy is saved by turning down a thermostat even by 1 degree? Is there a statistic available?" a Q&A answered by Michael Holcomb.