Warning: This thread contains companions' names and that can be considered a minor spoiler.
When Durance uses his staff and hit enemies, the combat log always shows that his hits result in "Burn damage". For example:
Durance hits Troll for 15 Burn damage.
And the red text of damage which pops up from enemies' heads are a bit higher, for example, 19 in this case.
However, after I enchanted a Fine Club, which is now being used by Edér, with Burning Lash, despite the weapon description gained a line of text of "+25% Burn Damage" like Durance's Staff and the weapon's visual appearance now has fire on it, the combat log still isn't showing that Edér's hits result in "Burn damage". For example:
Edér hits Troll for 3 Crush damage.
That said, I do notice that the red text of damage which pops up from enemies' heads becomes a bit higher, for example, 5 in this case.
Was the enchanting successful?