"Counter counters all physical moves for twice the damage done to the user."
From my understanding, yes it does. Say, your Pikachu got burned. It used Volt Tackle on a Wobbufet and it did 100 damage. That means when it gets countered, it will do 200 damage to the Pikachu. The trainer used a burn heal on the Pikachu, and it is no longer burned. It used Volt Tackle again, and the attack did 200 damage. If Wobbuffet used Counter, it would do 400 damage.
In short, Counter doubles the amount of damage you deal physically, regardless of burn.