Heat energy can be transmitted in three ways: Convection, conduction, and radiation.

Any properly-drafting fireplace will not output convection heat into a room. Proper construction practices prevent conduction to the inside surfaces. Radiation is the warmth we all enjoy from a fire.

A plasma or other flat-panel screen will probably generate more heat than it would receive from a fireplace. I'd be more worried about heat from a nearby air register aimed at the monitor.

Larry
www.fineelectricco.com