I think the burning is from vents at back of unit,

Yes, I would check your venting. Shut it down and open the access to the back of the fridge. The venting should be apparent and clean of course. Propane has a high water vapor content and contains Carbon Monoxide and should not be burned unvented in small spaces. Good Luck

I suspect the armature is shorted.

probably both as the arcing of the faulty brushes increase the current beyond what the insulation of the armature and field wires can cover

Probable armature burning , check blower brushes

A defect armature that happends oft by this tool multimaster there was a little serie of this 250 watt budget machines that have hiddens defects in the armature if you look at that part then you see not a burned field or armature but the tool becomes very hot to hold in your hands. if you are by a service dealer he can confirm the problem that in most cases not was your fault because if its not burned , but when its burned then replace BOTH the electric field and the armature if you can become that cheap , most times this budget tool is not to expensive to get new. but in that case the 350Watt tool little difference but more to do with.

You will rarely get smoke from the brushes on electric tools. Smoke is usually a result of either the armature or field coils getting hot enough to melt or burn the varnish insulation off the wire which results in a loss of or no power because of shorting in the windings. You'll have to replace whichever part burned the windings and find out what caused the problem or you'll burn the new part the same way the original burned.

You need to ventilate this area while the units running! This is potentially a lethal situation!
You are competing with a flame for oxygen and breathing the carbon monoxide gasses
from the heater! Both unacceptable scenarios! you can easily die in this environment!
I cannot stress this enough! You need to open a window or crack the door enough to allow air to circulte and gasses to vent out and escape, If the burning of eyes is really bad your unit may not have sufficient combustion air, I can t tell you without seeing it precisely, you better vent that room though!

Gas or electric?

1) Electric heater can be completely covered with insulation except TP valve. Set electric heater on insulated surface. Insulate pipes completely right down to tank.

2) Gas heater is different.
No insulation over combustion area, or gas valve, or air inlet or. No insulation on top of tank that will interfere with venting of dangerous combustion products including odorless CO gas. Foam insulation on pipes should stop 10" short of tank top. No insulation within 6" of the 3-4" inch galvanized vent pipe.

Here's photo with gas water heater insulation tips:
http://waterheatertimer.org/images/Insulate-gas-water-heater.jpg
http://waterheatertimer.org/9-ways-to-save-with-water-heater.html

3) Tankless gas. No insulation on box. No pipe insulation within 12". No insulation near vent pipe.

4) How much clearance between heater and chimney?? Not sure what that means. Is that the water heater vent stack? Or where vent stack goes into chimney stack? No insulation on gas flue pipes. Or within 6 inches of 3-4" water heater vent pipes

5) Is insulation fireproof? Everything burns if it's hot enough. If you throw insulation into backyard fire, paper and foil will burn away and insulation kind of falls apart in the heat. Fiberglass insulation cannot be set on fire with a match. Foam pipe insulation will burn.

It all depends on what is burning I'm afraid. When in use, run your hand around the back of the machine and handle area to see if anywhere in particular is getting hot. On the older versions of these (DW988) the leads leading from the switch to the brush holders would overheat and burn, also causing damage to the brush holders themselves. Is there any smoke coming from the rear vents, if so you wil probably find it is a motor problem and the smell would be the insulation burning off the armature. The only other thing to check would be to make sure the battery isn't shorting and getting hot, as this could also cause damage to the switch and vice versa.

Check the carbon motor brushes, there might be arcing on the armature. New brushes may help if the armature is ok.