There are no reasons that I know of for not filtering the air running through your stove. I have seen units where people adapted a sheet metal frame and enclosure to be able to slide the filter in.
When you install it just make sure that you check the filter regularly so you are not running it clogged. That would shorten the motor life.
Post a pic on here when you get it installed, I am wondering why stove makers don't put in some kind of filtration.