Assuming it's the nearest town and someone in Paulville begs them to do so at any price, yes. And then you hand them a fat bill and if they don't pay, enforce it via the court, something you seem not to have considered.
I am in no way suggesting that someone who is too cheap to buy a reasonable fee like $75 should get bailed out when they suddenly want in because their house is on fire, that's nothing more than rewarding irresponsibility. But it is not the only alternative to letting the house burn, a false dichotomy if ever I saw one.
There are two reasons to require that the FD respond if they can reasonably do so. One is economic; the limited damage and bill for extinguishing a fire might total to a $15,000 loss for the homeowner. If household earnings are $30,000/yr that's 6 months of work to pay it all back. But if the house burns to the ground, the loss is much higher; even in rural TN it's probably at least $75,000 up in smoke. That's a much bigger loss to the economy. Visualize it as a big bunch of dollar bills laid out along the ground, that starts burning from one end. I'm standing there with a bucket of water and you're offering me $5000 to put out the fire, which will leave the owner with 80% of the total, but I refuse and let it all burn because I'm still upset about your prior refusal to put down $75. Does that seem like a sensible policy? Am I somehow better off by your loss?
The other reason is legalist. The fire department has an effective monopoly on putting out fires, because only they have the necessary trucks and expertise. The homeowner didn't have the option of calling a competing fire department the way you would call a different cab company if the first one was unhelpful. Nor could one easily put out the fire oneself. In such a case, there's often a greater legal obligation to assist than usual.
Here, the fire department's non-action resulted in a much greater economic loss to the homeowner than if they had turned up and billed him. And they could have billed him, because a principle called quantum meruit means you can get a court to enforce payment for providing a necessary service even if no contract exists. Arguably, that extra economic loss, which only the fire department could have prevented, was intentionally inflicted. You can bet a case like that will go to court, and make the whole situation even more expensive.