Should we turn our oil hot water heater off every night?
Electric water heaters lose very little heat.  Putting a blanket on them or putting them on a timer won't save much money.

NG/propane/oil water heaters, on the other hand, lose significantly more heat, as they not only lose heat through their outside surfaces, but also even more heat via the flue.  Putting a blanket around the outside will prevent perhaps 10% of the standby losses -- worth it if you're using propane but not NG.

Turning the water heater off will lower the temperature difference between the water inside the tank and the air outside the tank, so it can save you a significant amount of money ($3-4 for propane; I'm not sure how oil compares to propane) if you go out of town for a week and turn the water heater off (because after a few days, the temperature difference for the disabled water heater will be nearly zero and standby losses, being proportional to the temperature difference, will also be nearly zero.)  Turning the water heater off for only 12 hours a day will probably only save you ~40 cents a week if you have propane.  Perhaps worth it if you can find a cheap device to do it for you, but not worth the trouble of doing it manually.

NOTE: By mentioning that lowering the temperature difference between water inside the tank vs. air outside the tank, one might be inclined to lower the setting on water heater to ~115 degrees Fahrenheit.  While this will indeed decrease your standby losses, this is not safe.  At lower temperatures, bacteria commonly known as "legionella" will grow, which is particularly dangerous to elderly and children, and sometimes causes death.  If you would like to lower the setting on your water heater, you must raise the temperature on the water heater to 135-140F for at least one day each month.