Huh o.o In the replay it shows that the burn goes and Lickilicky is cured ?Apparently Heal Bell does not work on its user, though it is supposed to:
http://replay.pokemonshowdown.com/randomdoublesbattle-177281348
Turn 5 Lickilicky gets burned.
Turn 6 it uses Heal Bell, but its burn stays.
EDIT: Fixed X)