In fighting fires there are essentially two strategies: remove the oxygen supply or reduce the amount of heat. Given that fire ants are not known for their low temperature, you most likely would not find success in any scheme that attempts to utilize the average temperature of a bucket full of the squirming red insectoid horrors to reduce the total energy of the system far enough to prevent rapid oxidization.
Given this, your best bet is to use them to dampen the oxygen around the combusting material. This requires an aforementioned large container full of fire ants to dump over the flaming hot thing, hopefully covering enough surface area to cause the fire to suffocate itself.
There are problems with this. First of all, fire ants are combustible as any obnoxious kid with a magnifying glass knows. Second, it is notoriously difficult to get the requisite fifty thousand ants to stay put in a bucket long enough to pour them over the fiery stuff.
All in all, probably too hard to be worthwhile. I would expect that your foot, or in a tight spot, cockroaches would be better.