I will be putting drywall back up on the pictured furring strips. The problem is, there is a ventilation shaft that goes below the furring strips, preventing the drywall from laying flush (previous ceiling had several layers, the last of which was popcorn texture, probably to hide the uneven surface). Elsewhere on this same ceiling, there is a waste pipe with the same problem. Can I lower the whole ceiling by installing another layer of furring strips (in the opposite direction?)? If so, screws or nails?
If it was just the ducting, I'd say go whack at it with a hammer until it bends upwards enough. However, you mentioned a waste pipe, which implies plumbing, which doesn't respond well to hammering. (Unless it's only hanging down by 3/8" or less, in which case I'd carve away the top side of the drywall.)
So, the simple answer is to cross strap on 16" centers. (Cheap 1x4 works well.)
The least fancy option is to simply nail it up without any regard for how uneven the existing strapping is. (You could screw as well if you wanted.)
The next fanciest option is to have a flat ceiling. This requires nails at the intersections so you can pull the strapping down to the point where it meets the string that you use to establish straight lines both down the strapping and across. After you've pulled the strapping down, insert shims and put another nail in.
The rolls-royce version is a flat and level ceiling. Again, nails on the cross-strapping, and you'll pull them down to meet the level of the lowest point in the room. Shim and nail as above. I can't express enough how useful it is to have a line laser and a story stick for this operation. (If you're putting cabinets into this room, flat and level might be worth it. If it's just an ordinary room, nobody would ever notice the level part.)