Aside from keeping the warm air from escaping, the doors act to keep the hearth clean just as a fireplace screen used to do in the olden days. With the doors closed, ashes and embers don't scatter about the room, keeping the carpet or rug clean and scorch-free. The doors also have built in devices that divert air towards the bottom of the fireplace. These vents make the fire burn brighter and easier to light.
Many furnaces are also equipped with glass doors that are especially necessary when the fireplace is warming up and can not deal with smoke and thus lets the bitter smell in the room. The glass cover helps to prevent it. The doors can be swinging or lifting. By the way, furnaces with a fireproof door are safer than open grates, and allow you to manage the flow of warm air much better and easier. The only drawback is that sometimes the heat-proof glass may be covered with soot.