If I understand an electric furnace, you need to keep the door to the closet open in order to allow the heat to get into the bedroom. If the electric furnace is making all the noise, and the closet door is open to allow the heat into the bedroom, then I must respectfully disagree with the previous advice about making the walls heavier.
Under normal circumstances, making the walls heavier by adding drywall would be a good gameplan to reduce noise transmission between rooms. In this case, I presume the door to the closet is open to allow heat to get into the bedroom, and that wide open door pretty well nullifies any other measures you take, including the double drywall thickness on the walls.
Here's why Slownsteady's advice would normally be correct for reducing noise between rooms. If there's anything you don't understand about the Mass Law, post your question and I'll try to explain it.
http://www.nrc-cnrc.gc.ca/eng/ibp/irc/bsi/85-sound-tranmission.html
The noise is almost certainly being caused by the housing of the electric furnace vibrating due to vibrations caused by the fan and motor inside it. I would try wrapping foam rubber around the electric furnace to dampen the movement of the metal panels of the machine into compression and expansion of the foam, thereby converting one mechanical motion into another rather than have it produce sound waves.
Just pay a visit to your local landfill and seek out old couches or camping mattresses. Remove the cushions from the couches and unzip them to liberate the foam rubber inside them. Also keep your eye out for plywood or any other kind of relatively heavy material that can be used as a new outer surface for the heater. Sandwich that foam rubber between the external panels of the electric furnace and the much heavier plywood or material you were able to obtain (even drywall), and hold evertything together with tape, rope, bungee cord, whatever.
Now, the vibrations of the light sheet metal panels will be dampened by the foam. Whatever vibrating the much heavier (plywood or whatever) external panels do will be of much lower frequency, and therefore won't be as audible to human hearing. The reason why is that if the panels are heavier, they have more inertia and simply cannot change their direction of motion as quickly as the much lighter sheet metal exterior panels of the electric furnace. It's really the inertia of the new heavy external panels that you provide that results in the sound deadening. Even if the foam rubber transmits the vibrations of the sheet metal panels through the foam rubber without any dampening, the heavier plywood or drywall panels have more inertia and cannot vibrate as quickly as the sheet metal could, and therefore cannot produce the higher tones that we perceive as being louder than lower frequency vibrations. But, the foam rubber will dampen the vibrations of the sheet metal exterior of the electric furnace, and therefore the heavier plywood or drywall panels should not move at all (or very little) resulting in little sound being produced by the machine.
Taking advantage of mass (and therefore inertia) to prevent vibrations and therefore reduce noise is one of the basic principles used by vibrations and accoustics engineers to reduce noise and vibration problems in buildings. The Mass Law of accoustics is explained in the link above. In a nutshell it says that noise transmission through a wall, floor or ceiling will be reduced by 6 decibels (or to 1/4 of the previous sound energy level) by each doubling of the frequency of the noise OR each doubling of the mass of the wall, floor or ceiling (per square foot). And, of course, the reason why is that a heavier wall, floor or ceiling simply can't vibrate at the higher frequencies that a lighter wall, floor or ceilng can because of inertia. If anyone is interested in the Mass Law, post again and I'll explain more completely.
Also, anything you can do to improve the balance of the electric motor and fan impeller to eliminate those vibrations in the first place will help the most.