Hi CT
I'm not an airport firefighter, but a friend of mine is, so I know a few things that they get up to on duty.
Im sure some of the more informed members here will be able to give you a more accurate response, but in the meantime.
At some of the smaller airfields / airports fire crews do undertake runway inspections. They will look for obvious hazards, such as obstructions, significant damage to the runway surface, they will also sometimes get involved with checking taxi and navigation lights and markers, may also be tasked to scare birds within the airport boundaries, gritting the runways in cold weather and snow clearance. Some firefighters will also be responsible for site maintenance.
They report back to ATC via radio and then normally sign an inspection schedule / log on return to station.
Not sure if fire crews do this at larger category / international airports, as generally larger airports will have specific technicians tasked with runway inspections, bird scaring, maintenance etc but from memory firefighters will normally assist with snow clearance.