A lot of people have a plasma over a fire, but it may reduce its life.
Open fires produce a layer of soot and chemical residue that sticks to the insides of modern electronics. This has an insulating property, as well as any direct damage that it may do, it may make the unit more sensitive to heat damage than it would at new.
Obviously heat is the main concern. If the electronics are trying to shed heat but the ambient temperature is higher than the component, it's going to fail at some point.
With all this considered, I helped a friend mount his plasma over his very nice fire. We did all the sensible things - large mantle piece etc, and tested the temp of the wall. The wall only got slightly warm and never hot. So up it went...
Cue fire on, cue plasma on. First thing that was noticed were circles of slight shadow effect on white or light backgrounds. They went away when the fire had not been lit for a day. Next time we felt the back of the plasma and it was a little too warm but nothing drastic. However, the metal mounting bracket was red hot in places.
The large metal bolts were so hot that they caused a slight distortion of the plastic around the mounting holes on the plasma. Although the wall was cool, the bricks underneath were very hot. Oops.
Despite the damage to the case, the plasma still works today, although it did not go back over the fire.