We had Radon Mitigation done in the crawl space and they told us not to put the vent covers in during the winter. The mitigation was a layer of plastic with the fan drawing air out from (I am assuming) underneath the plastic... Does this make sense not to cover the vents? Will I lose too much heat during the winter?

You may lose more heat if you do cover the vents. The mitigation fan needs makeup air to replace what it is pulling from your crawl. If you block the vents the fan may start pulling heated air from your home. If the fan can't get enough makeup air it will not vent the gases from under the plastic. I'm hoping that the plastic is sealed to the foundation around the perimeter.

I would recommend following the radon mitigation contractor's directions. A little chill is worth a whole bunch of "peace of mind".