The Goblin Fire Drum has an ever increasing ability to make explody things go BOOM.
Fire Drum:
All unattended flammable alchemical objects such as tindertwigs, flasks of oil or alchemist’s fire, and fireworks within this range have a 10% chance of spontaneously igniting or exploding (as appropriate to the item) when the drum activates. This chance increases by 5% each round and is checked at the start of the performer’s turn as long as the playing continues.
Would this work on a gunslinger's alchemical cartridges? And, if so, what would happen?
Yes, alchemical cartridges contain gunpowder, so they'd ignite. They'd probably explode on the spot and not much else would happen other than they'd be destroyed. Note that the fire drum only affects unattended objects, so alchemical cartridge's that are in a gunslinger's inventory would be safe. So long as he doesn't put them down they won't be subject to the drum's effect.