The fire, and police service really, are both based upon the idea that people are going to do stupid shit. More often than not (probably more so in the fire service than police), we are responding to fix your fuckup.
The scenario here is that a city's fire department basically contracts with homeowners outside of the city's corporate limits to provide them with fire service, because those homeowners live in an unincorporated area that does not have it's own fire protection. In the most technical sense of the scenario, it's not really their problem. It is not in their jurisdiction, and they do not receive tax money in the form of property tax from that residence.
In the me being a volunteer firefighter for 12 years sense, don't be a dick. While this was a mobile home and was more than likely totaled when you got there, show a little compassion. I agree they should pay for fire protection just like the guy down the road. But look, you bothered going out there in the first place to watch it burn and possibly to prevent any exposure to a paying customer's residence. So, if you took the time to go out there and be dicks, I'm sure the homeowner would be more than willing to then pay the $75 to have you put the fire out, as futile as it may be. Granted, he should have paid the $75 at the beginning of the year, but you went out there anyway, so let's come away from this looking good and not like a bunch of dicks on the news.
The only problem there is it opens the door for everybody to say "Fuck, I'm not going to pay at the beginning of the year. I will just wait to see if I need them first." The whole point of it from the city's perspective is to bring in revenue on the wager that they won't have to provide services to you. Then you have to worry about billing after services have been provided, the city is more than likely carrying a few more firefighters on payroll in anticipation of running calls in that area without money coming in to balance that out, etc etc. I can see it both ways.
But I also believe in the human side of what we do. I just couldn't imagine telling somebody "I'm not going to help you." For that matter, we don't even bill a resident's insurance company when we run a structure fire. If it's a commercial building, we will bill their insurance though. We bill them for apparatus, equipment, and personnel according to the rate schedule FEMA uses for fire service during natural disasters, but we cap it at $1,000 max.