That would be the flue control. The flue is the metal duct work, ceramic piping or brick chimney stack that runs from a firebox to the top of a chimney to the outdoors. The flue directs exhaust gases and debris from a fireplace upward to vent outdoors into the atmosphere. The flue is effective because it makes use of the stack effect that happens when there's a significant temperature differential between the indoor and outdoor ends of the flue. This has the effect of drawing the exhaust gases and ash, which are lighter weight than the cold outdoor air, up through the flue and venting them outside. Some heat is lost as well in this process, and modifications can sometimes be made in or above the firebox to minimise this ancillary heat loss.
The flue has a secondary purpose of allowing fresh air to enter it, which aids in feeding or oxygenating a fire. At the fireplace end that knob allows exhaust gases to escape, and fresh air to enter, when the fireplace is in operation. The lever can be closed to prevent cold air from entering a home when the fireplace is not in use.