The home side drew level six minutes before half time when Alex Goma brought down Clark in the box and Burns scored from the resulting spot-kick.
Steve Marlet regained the lead for Fulham on the hour mark but Burns grabbed a second equaliser for the home side in the dying seconds of the game.
Fulham: Van der Sar, Rehman, Goma, Djetou, Bonnissel, Malbranque, Clark, Inamoto, Legwinski, Saha, Marlet