I don't even know why the contractor put in the insulation. I would treat the utility room as conditioned space and insulate the outside wall, though not with fiberglass. Unless both the heater and furnace are completely sealed units with air supply and exhaust outside the home, there are mandatory provisions for ventilation. I'd check the manufacturer's booklets for specs. Exposed fibreglass is not a health hazard, but if the fg is disturbed it will irritate most people.

The town building inspector made us install a make-up air system even before we started any work, so we are covered ventilation-wise. The walls behind the furnace and heater are still original concrete, do you suggest insulation there? Also, any ideas on what to do with the insulation between the finished and unfinished space. Should I remove it?

The walls behind the furnace and heater are still original concrete, do you suggest insulation there?

Yes, extruded polystyrene covered by drywall. But if there's no space to work, at least try to seal the foundation to wood interface with caulking or polyurethane spray.

The fg can't hurt and might do a tiny bit to reduce sound transfer.