Does shutting off my gas fire when I'm away from it for a short while save energy?

When you leave the house, then YES turn off the gas for safety.

If you’re someplace in the house, but not in the room where the fire is, no need to turn off the gas. There will be some energy savings, but infinitesimally small.

Is it your basic source of heat? Then, no. It takes as much to heat the house back up again. If it isn’t your basic source of heat, your furnace will run a little bit more to make up the difference of no heat while the gas is off. In the end, the overall cost to heat will be exactly the same.

@kritiper No, not at all my basic source of heat. Just an auxiliary for coziness. In fact if I run it too long, the room gets too hot.

@janbb If you want to save energy, hook the fireplace through a thermostat so the room doesn’t get too warm. It will automatically regulate itself. (The thermostat would probably pay for itself in short order.)

@kritiper Thanks for the suggestion but I don’t think I leave it on long enough to need that. I turn it off manually when it gets too hot.

I would say that since you are using the fire for comfort in one spot, turning it off even for 15 minutes when you’re not there will tend to save energy, and would be worth doing. Particularly since, as you write, the room gets too hot if you run it for too long.

I disagree with kritper that you or the thermostat trying to manage the heat of the rest of the house would exactly result in the same amount of energy use. The heat system of a whole house is more complicated than that, particularly with two different heat sources, and your behavior and use in your case is about being cozy in one spot at certain times.

@Zaku I never implied, or meant to imply that the thermostat in that room would “manage the heat of the rest of the house.” I only meant that one room.
Any heat generated in that one room would, sooner or later, affect the heat in the rest of the house, and the thermostat that controlled the whole building’s main heat source.

For the most part a BTU is a BTU. I’m willing to bet the efficiency ratings on your gas furnace and the gas fireplace are very close. 95–97%.
The only way you are wasting energy is if the room gets too hot. Heat loss through a wall is directly proportional to the temperature difference between the inside and the outside surfaces. Ideally you want to keep the room at a comfortable temperature and no higher.
So, if you run it for a while and turn it off when you are comfy, and back on again when your flippers are frosty there is virtually no waste.
Go ahead. Enjoy!

Thank you for warming my heart with this question.

Is the fire open to the room or sealed behind glass? Because the first sends warm room air up the chimney and the second does not.

@Call_Me_Jay My installer said I had to keep the doors open all the time when I had the fire going.

@kritiper “I never implied, or meant to imply that the thermostat in that room would “manage the heat of the rest of the house.” I only meant that one room.”
– Was I wrong to infer that from you writing, “It takes as much to heat the house back up again.”?

In any case, it seems clear to me that janbb is using the gas heater for comfort in that one spot. And it’s a different source of heat (presumably with different efficiency) for the house than the system that heats the house, and there are various themodynamic effects of heat being in one place or another, leaking, etc, so for both reasons, I disagree with your statement that ” In the end, the overall cost to heat will be exactly the same.”