I'm not sure - I grow ordinary lilies and always leave the stems in place until they fall or snap off on their own, in the belief the bulb is building up food stocks for next year while the leaves and stem are functioning, but I've never grown a lily tree - as far as I can tell, they're just larger versions of ordinary lilies, so I'd be inclined to do the same with them.

I understand they are treated the same. so cut down the dead stalks from last year.