I don't think there's that much research. It's an allergen, which would point to an immune system weakening effect. The problem is that any research we have is probably on solvent-based whole-plant extracts. It does promote neurogenisis and cancer apoptosis. Also, given Adi's anecdote about the elderly gentleman in Hawaii who achieved lasting arthritis relief from kava, it seems to have a curcumin-like effect against inflammation.
One factor that must be considered in discussing the immune system is the impact of stress. If we give happy, optimistic, well-adjusted people who sleep 8.5 hours a night a pint of kava every night (4, 4oz shells), their immune system has an extra load on it. If we give a neurotic, insomniac, overloaded poor old sod a pint of kava every night, his stress level goes down and sleep quality and duration improves, so his immune function goes up. I don't know if it works that way, of course, but I'd be willing to participate as one of the gallon-per-day study participants.
Looks like someone else had the same idea.
This product is a mix of herbs (and zinc) that seems to be designed to relax the subject so that they stop fretting away their immunity.