Addicks striker Simon Makienok missed the two best chances of the first half, heading crosses from Morgan Fox and Cristian Ceballos over while unmarked.
Forest's David Vaughan shot just over the bar from 20 yards after the break.
Goalkeeper Dorus de Vries then kept Charlton at bay, tipping Makienok's header onto the post and saving Patrick Bauer's close-range effort.
Charlton, who started the season with a 2-0 win over QPR, have drawn their past two games but could consider themselves unlucky not to come away from the City Ground with all three points.
The hosts started the brighter before Charlton's giant Danish forward Makienok, on loan from Palermo, twice nodded off target in the space of four minutes.
After De Vries' two outstanding saves in the second half, Forest pushed for a winner but were restricted to half-chances as Henri Lansbury went close with a curling 25-yard strike and Jorge Grant blazed over from six yards.
Forest winger Michail Antonio was not included in his side's matchday squad and manager Dougie Freedman confirmed after the match that a bid from Derby had been rejected.
Nottingham Forest boss Dougie Freedman on teenagers Tyler Walker, Jorge Grant and Oliver Burke playing:
"They can see what we are trying to do. They had spirit, they had energy and there were loads of positives.
"These senior players are leading by example, and the young players are following in their footsteps.
"These young players are only going to get better with his experience.
"If Forest fans see their young players, their own players running around, they will love it. These players are only going to get better."
Charlton boss Guy Luzon:
"Their keeper is the man of the match without question. His saves showed how much we dominated; how well we played.
"I do feel that we should have won. We are disappointed because we had some great chances to score. We came to a tough place and played some very positive football.
"Our organisation was good, we pressed well and we generally did well. We had the chances to win the game and we are upset that we didn't take them."
Charlton maintained their unbeaten start in the Championship with a goalless draw at Nottingham Forest.