Do glass doors on a fireplace reduce the amount of warm air that get wasted up chimney?

When a fireplace is not being used... glass doors are not a good way to stop cold air from exiting the chimney. The first thing that needs to be done is the fireplace damper should be closed to prevent this heat loss. If the damper is not-functional or absent a chimney balloon or chimney top damper can be installed to stop the heat loss.

When there is a fire in the fireplace...Fireplace glass doors reduce the amount of already heated interior air that the fireplace has access to. So it does slow down the inside heated air from being cast up the chimney with the fires combustion heat, but glass doors do not stop this heat waste completely. Fireplace glass doors also inhibit the only usable heat the a fireplace produces (radiant heat) from coming into the living area. Standard open fireplaces and fireplaces with glass doors are notoriously inefficient at heating the home. According to the US Department of Energy using a fireplace in the winter will actually make your heating bill go up by 10% on average.