It's always best to shut the vehicle off before refueling.

Fuel is flammable. Cars generate a lot of heat. One spark is all it takes.

That's why gas stations have signs that say, "Turn off motor before refueling."

Aside from danger of fire, most modern cars will trip the check engine light on if the gas cap is loose or removed with the engine running since that's detected as a leak in the vapor recovery system.

Listen to @mod_man. I don’t know about you, but I never want to find out the hard way..

Check local/state regs. I’d bet it’s required 

"Is it best to shut the engine off before refueling?"

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Yes. This would be, … just perhaps, the worst possible method to groom your eyebrows, trim eye lashes, … and square off bangs!

F.S.

It’s always a good idea to turn off the engine before refueling.

You are asking about hot refueling. It is common in military aviation. Static electricity must be eliminated to prevent possibility of sparks that can ignite fuel.