I might add that the primarily "fire rating" of wall and floor/ceiling assemblies is to allow occupants to safely exit the building, not to keep the building from burning down. The Type X gyp board affords some protection because the inherent moisture in the product can slow down the fire through your wall or ceiling. With these assemblies, you are basically looking at a way to escape the situation before your obvious portal to Hades has the structure tumbling upon you.
Perhaps you should also look into a fire suppression system--not sure what the initial source of flame is (what might be burning other than building components), but sprinkler heads tied to a detector might be a good idea.
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.
Best Answer
We often design and build “safe rooms” in custom homes. Safe rooms have several requirements: 1) structurally secure, 2) fire protection, 3) temperature and humidity control,
You are requesting information about fire protection only. However, it might be important to have a space that can protect you structurally in case of a severe wildfire AND keep documents in good condition year round.
Structurally, reinforced concrete block is an excellent choice for walls, but you’ll need a suitable ceiling. I’d recommend a “heavy timber” ceiling covered with several layers of gypsum board or a structural concrete slab.
Fire protection is based on “hours of protection”. That is to say, all materials are given a fire rating depending how long you need to keep a space safe. The Code requires a 1-hour rating between the garage and a home, which is about how long it takes to get everyone and everything important out safely. I’d recommend 4-hour protection, but you can’t obtain a 4-hour rating with wood framing. (You’ll need concrete block and/or concrete.)
You can buy fire doors that have fire ratings too.
Floor construction is less important, but using a non-combustible material like a 4” concrete slab on grade would be appropriate. (Make sure ground water and water table is accounted for and the area is adequately drained with a french drain.)
Depending on where you live, access to fire protection, etc. will help you decide how secure and how much fire resistance is required.