Drywall – the code approved way to cut and patch fire rated drywall hung on the ceiling of a basement

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.

Codes are usually minimum safety standards.

However, in the vernacular, "code" is a loose term that includes national, state, county and local laws and ordinances that dictate or direct building practices of all kinds. The whim of the inspectors could also loosely be considered "code". Most of the time, the basis of these laws and whims are guided by or reference/include the works of standards bodies, such as the National Electrical Code, NFPA, and International Plumbing Code, to name a few.

Depending on the jurisdiction and disposition of the building inspectors, a good way to make sure you will meet code for a particular project is to ask the inspector to look at your plans before you start your work. Your mileage here will vary greatly. Some jurisdictions won't even return your calls if you're not a contractor.

A good way to plan a project is to find a book at your local hardware store on the subject. While "code" varies from place to place, there is a lot of uniformity and common ground. These practices will generally be followed in these books. In fact, many times these books address code directly. A book I recently used on deck building referenced the International Residential Code specifically in many sections when explaining a decision to be made.