How cool should a firebed be to allow closing the flue?

By flue, I believe you mean the damper in the chimney, which has the effect of closing off the flue. Close the damper any time you are not using the fireplace. Remember -heat rises -so heated air will also rise up and out the flue in the winter. In the summer it is also good to keep the flue closed because heat and humidity will activate the smelly properties of the ash and creosote in the firepalce chimney and create a bad odor in the home. In order for a damper to do its job it must seal tightly/ If your existing mechanical damper is broken, missing or in a bad state you can repair it or replace it with a chimney top damper or a chimney balloon. But, open the damper before lighting a a fire, or it will rapidly get quite smokey in the house.