I looked up research about specific interactions of kava with aspirin, acetaminophen and ibuprofen. Here's what I found.
Acetaminophen
It should be noted that there a couple studies about the specific interaction between kava and acetaminophen.
1. One study in-vivo suggested that kava might exacerbate the damage to liver cells caused by acetaminophen. The study has some issues and is not definitive: it used an unknown type of kava extract, not traditionally prepared kava, and involved exposing non-human liver cells in a test tube, probably to much higher doses of both acetaminophen and kava extract than anyone would normally consume.
Kava extract, an herbal alternative for anxiety relief, potentiates acetaminophen-induced cytotoxicity in rat hepatic cells
2. There was another in-vivo study in mice that showed that kava extract (and also the FKA and FKB alone) potentiated liver damage caused by acetaminophen (although kava extract by itself showed no toxicity). Some of the same caveats as above apply: the kava was an ethanol extract, not traditional water extract; the subjects were not human; the doses were much higher than people would normally consume (500 mg of extract per kg of body weight):
Flavokawains A and B in Kava, Not Dihydromethysticin, Potentiate Acetaminophen-Induced Hepatotoxicity in C57BL/6 Mice
So both of these studies do have some obvious things about them that could be criticized, as noted above. Nonetheless, they are specific results about a specific interaction, not a generic warning about anything that is metabolized by the same liver enzymes as kava. Since they are the only solid information we do have, in my opinion the usual advice to avoid combining kava and acetaminophen is probably wise.