You always want to let lily stems and foliage die back naturally. Like other bulbs, they need all that 'green' to remain in place as long as possible to photosynthesize and recharge the bulbs for next year's flowers. Once they dry up and turn brown - usually late summer or early fall - you can very gently dig up the bulbs and separate and replant. You just need to be careful not to damage the bulbs when digging.

Thanks so much! Just the information I needed!