As best I know, a fire rated utility room isn't required by any code for a single family structure. And for multi-family (e.g. condos), this requirement is to isolate each residence from each other and from common/utility areas. So the below advice is completely overkill.

For the room itself, you can use fire rated drywall. This is usually 5/8" thick, and contains fiberglass within the wall to keep its structure during a fire. Use this on all walls and ceiling. Around any protrusions, (HVAC vents, water pipes, etc), you can fill the gaps with fire rated expanding foam (e.g. Great Stuff). This is usually pink/red in color instead of the normal yellow. For HVAC vents, you can get spring loaded dampers that automatically shut in the event of a fire to prevent it from spreading via the ducts. You can get a fire rated solid door, but to do so, you also need to get an outside air source into this room.

All of the fire rated products are to slow the spread of fire, not to stop it completely. The goal is to get you out of the home safely, before it spreads and you're trapped. Therefore, I'd recommend installing a fire alarm that is connected to the other alarms in the house, ensuring that if a fire starts in this room, you'd know about while there's plenty of time to escape. A standard fire alarm in an enclosed room could easily go unnoticed, or allow you to sleep through it.

If condensation can be ruled out, and it only leaks when it rains, is it oil or gas fired? If yes, then it most likely has a chimney, and probably a metal one. Depending on your roofing material you should have someone look at the flashing around the chimney.