Well, when the Y2K insanity was quite ripe, my wife actually encouraged me to get a generator. It sat in the garage for some time, then we had a nasty ice storm. I was in the kitchen taking my evening pills and listening to the "snap, snap, THUD" of assorted branches going down, and it was really going like as fast as you read that! Finally, there was this huge "BZZZORCH - BZZORCH -BZZORCH" as the recloser on our branch cycled a couple times and shut down the line. (We have 7200 V residential distribution here.) A tree a couple houses down had leaned over against the feeder.
So, I went down in the basement by flashlight and disconnected the furnace from the permanent wiring and put a standard wall plug on it. Then I went and strung extension cords through the house, and went to bed, as it was still warm. My wife woke me up around 6 AM and said "it's getting awfully cold in here." So, I went out to the garage, opened the garage door manually and rolled the generator out to the open doorway and pulled the cord. I plugged the extension cords in, and the furnace, refrigerator and freezer fired up. I let it run about 90 minutes until all appliances were happy with their temperatures, and shut it down. I was just about ready to do it again at 4 PM when the power came back on.
Unless your furnace had a 12 V (or whatever) backup battery to run the controls, and maybe a generator to keep the battery charged off the gas turbine, it wouldn't work anyway. At least as a heating system, efficiency of the gas turbine would not be a concern, it would just add more heat. I'll bet some industrial or military equipment maker already makes some similar apparatus, I know aircraft preheaters pretty much meet this definition. Don't WANT to know what those cost!
Jon