For a few years I learned how to fight with medieval weapons. Some people I met then had their swords put on fire (in this case, with oil on a wick, so it was for a short time)
It was beautiful and impressive ! But not very effective.
Of course they did not use it to fight an army for glory. It was a choregraphic fight and the "enemy" should not be hurt or frightened. The public was impressed, the noise of the flames and the light were spectacular, but the fight itself has to be quite slow and simple.
The flames are as dangerous for the owner of the sword than they are for the opponent. If you want to protect yourself in a swordfight, you have to keep your sword very close to you. Keep your hair tied and protected, and make sure your clothes won't ignite. And you will have to wear good gloves, as your wrists will stay very close from the heat during the whole fight.
As long as you move quickly there is no problem, neither for you or your opponent (as it has been said, the heat don't hurt you if you don't let it on your body for a moment), but in a real swordfight your swords will meet, and stop, and then move again. Your opponent's moves will bring your own sword to protect you, thus put it very close of your head, your eyes, and you don't want flames near your eyes.
Perhaps, if you manage to hurt your opponent with your flaming sword, it will hurt him more than a regular sword. Burns are very painful, but so are "regular" sword injuries. If you manage to hurt your opponent, a flame sword will be a little more cruel, (burns are painful and take a long time to heal) but it won't help you win the fight, except if your opponent is terrified by your mighty flaming sword and run away as soon as he sees it.