Automatic flue dampers are found on high-recovery commercial water heaters. There usually is a large butterfly valve in the exhaust orifice on top of the water heater (shown in the yellow circle) connected to a small electric motor mounted on the cover (red circle). When the thermostat signals that the tank needs to fire, the motor opens the damper as the electronic ignition fires to start the water heating cycle. They are required by law as an energy-saving device.

That said, the devices are finicky and not everybody is qualified to work on them, so there are often problems with them.