If you set your refrigerator to "auto" it will automatically switch from electric to gas if the electricity goes off, or if the trailer is unplugged.
To use is effectively, you do need to leave your propane tank turned on.
It does not hurt to run the refrigerator 24x7x365. We keep ours on all year, except during the coldest months of the year, when the outside temperature is actually colder than the refrigerator will get. Then the refrigerator won't turn on at all, meaning the freezer won't freeze, unless sub-zero temps penetrate the inside of the refrigerator.
If you leave your refrigerator on 24x7, it's best if you leave something in the freezer compartment and the refrigerator compartment. We use the trailer refrigerator as a second refrigerator at home. (since it's parked at home all the time anyway). So there is always something in the freezer and the refrigerator compartments. We usually keep drinks in the refrigerator and frozen meats in the freezer. Usually after camping we have an abundance, as we usually pack food pretty heavy! So all week long, we are taking stuff out of the camper refrigerator, and then restocking for the week-end camp out again on Friday. This also keep everything circulating, and nothing gets old.
But to answer your question, use gas only ... well ... actually, you never need to use "gas" only as "Auto" will flip to "gas" automatically. Use "Auto" for electric and as a back-up if power goes out. If you keep food in the refrigerator this is really a good safety net so anything in the refrigerator won't get warmed up or thawed out.
"Off" turns everything off! When turning "Off", make sure your refrigerator and freezer door compartments are propped open so they don't get smelly.