I really have no experience running with a closed chassis, but just some thoughts...
A fan would help circulate air around the chassis, even if you left it closed up.
Better idea: I would put a few holes in the body in front of and behind the motor to allow air flow through the chassis and across the motor. Then, go get some fine mesh screen and shoo-goo it across the openings, inside the body. Belt drives aren't too susceptible to fine dust in the short term; the real fear is small rocks, clumps of dirt, pebbles, etc. So the fine mesh will keep those out while allowing airflow. Its kind of a compromise, as you'll have to clean the belts and pulleys kind of frequently (because of the dust allowed in), but you won't risk insto-death for the belts and will have a cooler-running car.
I don't think there are really any competition off-road cars that are sealed up by the body, for the reason you're pointing out. Even cars such as the 501X and the Yokomo BX, which have open belt drives, have large cooling holes for the motor and electronics:
Hope that helps some.