Here is a link that I did for a Dust Collector and within that link is a link to a "box" that quietens the DC system. One thing that I learned was that the best sound dampening comes when ALL sections including INLET AND OUTLET edges for the box's air flow are covered.
http://www.penturners.org/forum/topic.asp?TOPIC_ID=33477
For you, I think you will need some kind of small fan to pull air in and push air out of the box to remove the heat. Do this in a baffled inlet and exhaust for the greatest noise reduction. In baffles and edges, I used a very thick felt carpet that construction people use to place on finished floors to walk over.
In placing carpet on the inside walls, I used mostly carpet squares and the easiest way to attach them were with 3M type of spray adhesives.
On the box that I made, (unlike an air compressor box) The air is exhausted into a small bag INSIDE the box in a chamber under the DC itself. The exhausting air flows out of the bag, up and over the DC unit and out the baffled exhaust, pulling all heat with it.
ADDED NOTE: IF you placed it in a box about 4 feet tall, with carpeted exhaust openings at the top - with say - the top raised 2 or 3 inches above the walls with corner braces of 2x2, all heat could escape without the need of a fan. If all inside surfaces were covered with carpet, the dampening effect would be considerable. IF you cover the outside too, it would cut down even more on noise.
Solid surfaces either _reflect_ or become a _sounding board_ increasing the noise effect, but totally covered, they cut down considerably.