Drywall – Ceiling drywall – options for dealing with joists slightly higher than beam

The way we do it is to strap the joists perpendicular with 1X3 cheap strapping. Find the lowest point in the existing field of joists and that will be your level reference. We use a laser 360 degree level , but you can get by with a good 6 foot standard bubble level. Mark off the joists in 16" on center with an extra piece of strapping along each wall edge. Then set the first piece of strapping closest to the lowest point of the ceiling. Use simple shims under the strapping as you go to keep each piece of strapping level side to side and row or strapping to row of strapping. Secure the strapping with 4d or 6d ring nails to the joists. It will take a little time, but you should get the hang of it very quickly. The result should be a nice flat and level ceiling. I know this is a quick description, but I will be happy to give you more info and hints if you decide to use this method. BTW, this is definitely a 2 or 3 person job, and an air nailer will really help.

This will likely be a location specific issue, so it doesn't hurt to check your local codes.

All the fire rated work that I've done has been between units (both to other units and common areas) and any load bearing structures of a multi-unit dwelling. When we do that, fire blocking is installed in the ceiling space as well to prevent a fire from traveling through the ceiling to another unit. For everything else within a single unit, and in a single family home, we don't use any fire rated drywall.

Edit: A few reasons I'd disagree with Eric on this one. First, I'm presuming you already have a certificate of occupancy (or someone did) to move in with an unfinished utility room. That required an inspection that presumably allowed the space as is. Next, most codes for combustion devices (furnace, gas hot water) require that you have ventilation. Installing a vented door is perfectly acceptable and won't provide any fire stopping. And, finally, most HVAC's (if the furnace is forced air) will be connected to vents that run to every room in the house. From the little I saw, the furnaces and utility rooms in searches likely referred to multi-unit dwellings, and that does need to be fire stopped. You also have to use fire caulk around every hole. Our multi-unit buildings also have sprinklers throughout and spring loaded vents that snap closed in event of a fire.

That all being said, there's absolutely nothing bad about installing fire rated material around the furnace, other than the extra cost and that it's a bit more difficult to cut. If you want to do it, then by all means, do so.

Edit 2: One last note, realize that fire rated drywall increases the resistance from 30 minutes that you'd typically see with 1/2" drywall to 1 hour. Fire rated doesn't mean fire proof.