The extra damage will not trigger if you are enchanted with Illusionary Weaponry even if the physical weapon you are wielding is a fire weapon.
You can use a Fieryupgrade prefix for your non-spellcasting weapon to enchant it with the element that Conjure Flame uses.
Any skill that causes your weapon to deal fire damage will trigger Conjure Flame, even without a Fiery upgrade - for example, Kindle Arrows or Conflagration.
Heroes will use this skill in combat even if the condition isn't met.