It Totally Safe for the Pilot Light of a Water Heater to Be Lit While the System Is Turned Off
The question of whether or not it is safe for the pilot light of your water heater to be on when the heater is not is a redundant one. You have no choice in this matter, as the manufacturer of your heater or water heater will probably dictate that the pilot light stays on when your heater is not on. It’s designed that way.
New pilotless heaters or water heaters have small flames which are automatically generated when it is needed. In most cases, the pilot light must stay on both for proper operation of the device and to prevent flashbacks that can occur if the pilot light goes out and you try a restart.