The best-known instance of pigs being used against elephants was the battle of Maleventum where the Romans "drove them [Pyrrhus's elephants] off using pigs coated with grease and pitch, which they turned into live torches. Their [the pigs'] shrill, dying squeals made the huge pachyderms panic and put into flight"

So, that's the bulk of the story, right there. The Romans set some pigs on fire and the elephants didn't like the sound and shied away. As would I. Healthy, happy pigs make horrible noises. Pigs burning to death must be about the worst sound there is. Also note that the story doesn't say the pigs charged the elephants. The Romans might have kept them in pens.

I don't know if being set on fire qualifies the pigs as combatants. It certainly doesn't make them "war pigs" in the sense of a "war horse."

The Romans--and many other cultures in the ancient world--were wary (or just frightened of) elephants in battle because the great beasts were the day's equivalent of the tank. Whenever some invading army showed up with elephants (Pyrrhus in this case) the Roman's lacked an easy counter (they had no elephants), so there are a number of Brer Fox-esque tales where the Romans defeat the elephants through trickery. Aside from the flaming pigs, the Romans also tried building these crazy calliope-carts filled with anti-elephant caltrops and specialized halberd-like devices designed to cut their trunks and flaming grappling hooks. I imagine them looking somewhere between a da Vinci drawing and something out of Mad Max.

The carts didn't work. Pyrrhus just kept his elephants away from them and sent archers to kill the men inside.

On an unrelated, but heartbreaking note, several primary sources also mention that female elephants with calves would refuse to fight (or just go straight crazy) if separated from their offspring, so sometimes those calves had to go into battle with their mothers. If those babies were harmed, the mother was likely to go on a rampage, often killing indiscriminately.

It's like Dumbo in hell.