Look up the type of furnace you have on line and see if they have a manual.
They usually have trouble shooting guides in the back. For the Basic stuff.
There is a lot of stuff that is not in the manuals though that sometimes comes with experience
I work on a lot of RayPac, American, A.O.Smith commercial water heaters. Not all are the same but many are similar in many ways.
My furnace is old and does not have a pilot safety. It has a standing pilot.
The pilot will stay on if I unplug it and will not automatically shut off if the flame goes out. NO thermocouple. Just to see if I am correct I turned off the gas a couple minutes ago. Yup Just stuck a lighter to it after I turned on gas and it relit. It does have a I think a thermopile that lets the controller/thermostat know it has a pilot flame before it will open the main burner.
works good.
It all depends on how old that heater in ? is. does it have a button on the gas valve that you have to push in to light the pilot manually or is it electronic ignition. If it electronic ignition then I agree with Frodo, If you turned power off and pilot was still on . probably a bad valve . If its a standing pilot that you have to light manually then the power has nothing to do with the pilot flame.
If it has a thermocouple or thermopile then look for a button to push to relight the flame some are in the on/off dial, some are seperate
