quote make the claim that they can reduce your bill by reducing the amount of gas your furnace burns or making your natural gas mix better
As I understand the device, that is not what it is doing. It is not changing the gas pressure or air valve or air to gas mixture in any way instead it is duty cycling the burner. The burner operates just as it did before only on and off. The end result is it is reducing the total capacity of the furnace and it will run longer. Therefore you should pick up some efficiency since your total run cycle will be longer and you should therefore have fewer cycles, furnaces waste energy warming up - they turn the burner on but wait with the fan to avoid cold drafts. The burner uses gas but you get no heat. Also with a duty cycle overall flue exhaust temps should be lower. OK On moderate days but you may have a problem on really cold days if it doesn;t somehow sense that you need full capacity again.
One thing you can do that's proven to work is to provide a fresh outside air supply to your burner. A electric damper can open it whenever the furnace is running, this will prevent the furnace from sucking in outside air under your doors etc. If it fails no problem you go back to what you have now as opposed to dampers on the exhaust flue. You can also check to see if you can switch up one speed on your blower if available.