Fireplaces, while first impressions lead people to think of warm and cozy thoughts, are in reality one of the least heat efficient items in a home.
These are infrared pictures of a fireplace opening with and without a door installed. As you can see the temperature readout on the fireplaces with doors can be double that of those without.
It is also helpful to note that fireplace doors are not intended to be closed during the fireplace operation. The main purpose behind doors is for heat efficiency while the fireplace is not in operation.