If you live in Georgia, then yes, you probably should if for no other reason than it appears to be the law. That’s probably true in a number of other states as well, but a quick review of law-enforcement officer bulletin boards suggests that many police don’t think it is illegal and don’t turn off the engines in their police cruisers while refueling.
It is an odd thing for people to be concerned about – how could running the engine in the presence of gas fumes lead to an explosion when starting the engine somehow doesn’t.
But whatever – there does appear to be a fair amount of public concern about possible explosions related to gas fumes being ignited by a running engine – just as there is a fair amount of concern about explosions triggered by cell phone use at gas stations. However, there has never been a documented case of a gas station explosion triggered by a cell phone, and I can’t find any evidence of an explosion or fire triggered by an engine left running while the car was refueled.
That’s not to say there are no dangers associated with refueling. Sparks from static electricity discharge appear to be a real concern. Over the past couple decades there have been hundreds of cases of fires and explosions caused by static electricity sparks – sometimes resulting in serious injury. But the more substantial danger of static electricity at the pump – for some reason – has failed to garner as much attention as a couple of concerns – running engines and cell phones – that are completely fanciful.