Trailing to Harriott's 10th minute goal for the visitors, Stuart Pearce's side responded well after half-time and it was substitute Robert Tesche who levelled the scores on 59 minutes.
The remainder of the game was steady traffic towards the Charlton goal but the Reds could not find the winner in the same way as their dramatic victory against Norwich the last time they were at home.
A point keeps them above their opponents and sees Forest lie 10th in the Championship table.