Can sheetrock be put in for fire barrier over existing ductwork and plumbing in drive under garage?
It looks like about the only thing you can do is add a drop ceiling in this area. Code here would require the drywall to be 5/8ths between a garage area and a living area. Basically this means putting in a frame to support that drywall. It wouldn't have to be finished except on the joints, and if you have an area you need to leave access to, you can make that a removable panel (leaving access to the screws that hold that drywall panel up, or creating a hinged door for it.) On the bonus side, it would hide all the plumbing, AC, etc that is now exposed. On the down side, yes, it's expensive, about $400 per 4X8 foot area to do this properly, with the frame-in, unless you do it yourself...
Part of the consideration for this is also fumes and not just fire, otherwise you could probably install a drop acoustic panel ceiling, which is much easier, and slightly less cost. 
Also, I'm surprised the inspection found that door to the stairway acceptable, it should be a 20-minute rated fire door, even if one is installed at the top of the stairway, with proper self-closing mechanism!