The best way to tell if a fireplace damper is open is to take a flashlight and actually look up the chimney from within the fireplace. If the damper is closed, all that is actually seen is an obstruction. If it is open, it is possible to see straight through to the sky.
Fireplace dampers are designed to be closed when the fireplace is not in use, thus keeping cold air from coming in and warm air from escaping. However, they absolutely need to be open before a fire is lit in the fireplace. A fireplace lit with the damper in place results in a house full of smoke. It is also important to make sure not to close the damper until ashes are completely cold. The reason is that carbon monoxide can build up in smoldering coals and ashes and present a silent danger if not allowed to escape. Gas fireplaces have dampers that are always open, as these present a significant carbon monoxide danger otherwise.
Dampers are not completely foolproof in terms of keeping heat in or cold out. One way to create a tighter seal is to install glass doors on a fireplace. With these shut, cold air is less likely to get in.