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.
A good drywall adhesive has much more shear strength than screws - and this is just when your glue is on the framing. So if you are doubling up the drywall gluing makes a ton of sense.
Just recently my drywall guys started using glue and I can say this, I hope I never have to demo these places because the drywall is ATTACHED.
All this being said the glue holds to the back paper of the drywall. What you are saying is that behind the drywall you don't have a solid base... then the glue won't work.
Also I am going to assume that crumbling drywall means moisture. While most glues will be fine with a little moisture, the back paper of your drywall will not.
Then we have to think about heavy tiles. Heavy tiles do not go on drywall. They go on a backer board.
You need to take out crumbling drywall, then install 1/2 inch Hardiboard, then install tiles. There is no reason in the world that you need exterior access to remove interior drywall.
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Best Answer
Sure you can. It will be a better water deterrent than painted drywall and will certainly be easier to maintain and clean.
Is it recommended? Probably not. But not everything has to be approved.
I would adhere Formica with the same mastic for applying vinyl tile. It is easy to apply and work, and has more than adequate adhesion.