So what is a barometric damper? That is the strange swinging plate you find somewhere between your oil furnace or boiler and the chimney. Its function is to allow just the right amount of air into the flue pipe as the gasses go by to let the chimney perform its best.
What goes wrong with a barometric damper? Some people play with them too much, like trying to adjust the little weight that determines just how much it opens. Don't touch that, let the furnace repairman take care of that during his annual tune-up. But you should walk by every once in a while and tap the plate to see that it swings freely. In the first photo you can see it stuck open and the furnace is not even on. It should only be open like this when the furnace is working.
What happens is that the two little pivot points get dirty, and it either blocks open or blocks closed. Just a little squirt of oil on both pivot pins and make it swing a bit and it will regain its freedom -- helping your oil fired furnace or boiler to work properly.