RAW doesn't sayProbably a DM call, but... there are some precedents
This is going to be a DM call either way, but the DM probably wants to use the torch and the improvised weapon rules to inform that ruling, so it can go one of several ways:
- +1 fire damage: The burning weapon is similar to a torch, so it should act as a torch on a hit as well as performing its own "job."
- 0 additional damage: Despite being on fire, the amount of flame that a burning weapon produces is likely far less than a torch, a weapon not designed to be lit aflame isn't as good at holding a fuel supply as a torch...as well as probably a number of other reasons mean that the heat from the flame isn't enough to do a significant amount of damage; it's more akin to jumping through a campfire than falling into a campfire. A flaming weapon like this should still be able to set highly flammable things on fire (like flaming arrows over the castle wall meant to set thatch aflame).
- Choose 1 or +0: Hitting someone with a flaming weapon might not leave the heat in place long enough to cause damage, but a weapon could be held up to a target, foregoing its normal effect in favor of acting like a torch in combat. You'd only really use this tactic if you were facing an enemy that could be damaged by fire but not by the weapon.
- Convert one damage to fire: The weapon is no better than before at hurting someone, but now at least part of its effect is the result of the flame, so out of a character's damage roll, the weapon deals one fire and the rest according to the weapon type.
...there are probably other ways to rule this; as with most DM rulings, the key is something that fits the theme of the game, that everyone can accept and that's fairly consistently applied throughout.