IMHO, I do not think you can cover with drywall. The pull down stairs I have seen (and installed) are designed to be flush with the existing ceiling. Also, the springs are designed for the weight of the stairs and the wood door. I suspect the additional weight would cause the door to not close properly.
You MAY be able to install a thin piece of sheet metal (stainless steel would be my preference, as thin as you can get it) to get some level of fire proofing. Aluminum would probably melt too soon.
Hope this helps.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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