I would assume the expense of ARFF is part of the Airports budget. If ARFF was funded by Fee-per-use, those firefighters would starve to death! and they would have 1960s apparatuses. But how many times have you taxi'd around some small airport and they have a new Stryker driving around the field? I see it all the time. Thats close a million dollar truck!
and EPP, I would hope you are wrong about people not declaring an emergency because they are afraid of a bill. but I can see how some people might be like that. Which is wrong.
If the airport is a part 139 airport, then ARFF is required be at the airport and no cost will be passed onto the pilot/company. If you are landing at a small field then the local fire department/EMT's will have to respond and it becomes a gray area. Worst case is a partial amount of the expense.
If you do declare an emergency, just be ready to do some paperwork on the ground. You'll have to explain what happened. If you land at New Orleans Int'l I'll be the one asking you those questions! I've had to interview several Captains on what happened and they get an attitude and think I'm questioning their judgment, but I'm just trying to get the information for the report I have to write. It's really not that big of a deal though, it happens at least once a week. The most common is the gear not down and locked indicator.