Yes remove the brown stem and all the leaves. Burn, throw in the trash or bury elsewhere covered with soil. The reason that this is important is to eliminate overwintering Botrytis on the dead plant debris that could infect your lilies next spring. This is a very common disease on lilies and Asiatic lilies are particularly sensitive to it, so it's well worth removing the dead stems as a precaution, even if your lilies haven't shown any symptoms yet.
