My understanding is that it procs when first activated... secondary activations, like Sion's stun or Janna's tornado, would not set it off. (On Lux, the proc activated when you first cast your Q, not when it detonates, which is the same idea as the above 2 champs.)
So abilities like corki or Ashe where multiple shots are fired, only activate it when you first fire the shot. For toggle abilities, they are a passive and from I have seen, do not trigger the ability in any way. On hit abilities also do not proc Lich Bane, so Teemo's toxic shot or his use of Lich Bane is extremely limited at best.