the LOTR Balrog is essentially a creature of fire. THus, the dragons fire attacks would be ineffective against such a creature.
However, (Unsuer on LOTR dragons) its generally assumed dragons have both physcial attacks (tail, bite and breath weapon) in addition to magical abilites. So, while the dragons breath weapon would be ineffective, dragons still possess mythical strength as does the balrog, and magical abilities which could offset the lack of an effective breath weapon.
Also, neither creature is dumb. So both would try to place the fight on grounds which would help them.
I'd say it comes down to where its fought. In a limited space (Cave, dungeon) where the dragon wouldnt fly and was restricted to "In your face" type of fighting, I'd give nod to the balrog. In a more open space where the dragon would fly and have time to set up and use magical abilities (assuming our dragon has them) I'd give nod to the dragon.
This is why who picks the fight is so important. A dragon most likely wouldnt march into the balrog's cave any more then the balrog would run out in the open to get picked apart.