Vincent Kompany's side soaked up some early pressure and then took the lead after 41 minutes with a superb finish by Zeki Amdouni after good work by Luca Kolesoho.
Burnley grew in confidence and looked comfortable in the opening stages of the second half but were rocked when Callum Hudson-Odoi curled in a superb 61st-minute equaliser.
Forest sought to maintain the momentum but it was Burnley who thought they had re-taken the lead when Lyle Foster slotted in Sander Berge's low cross but the goal was disallowed for a harsh-looking handball against Berge.
There was worse to come for Foster who was sent off in stoppage time for an elbow.
The draw lifted Forest to eighth place in the table with seven points from their five games while Burnley are 19th with one point from their four games.
Burnley will feel they should be celebrating their first win as the decision to disallow Foster's goal looked harsh, with referee Robert Jones deciding to rule it out after being invited by VAR to look at the pitch-side monitor.
Berge had brushed off Forest defender Scott McKenna all too easily before cutting the ball back for Foster but his arm has brushed against the ball as he raced into the area.
"Even the defender who was in the duel with Sander didn't see anything," Kompany told Sky Sports as he watched back the incident in slow motion.
"What do you want me to say? I really can't do anything about it, I don't feel anger I'm just proud of the team and the squad right now."
Forest began the game full of confidence after their impressive start to the season and pinned Burnley back in the opening stages with Hudson-Odoi forcing a save from visiting keeper James Trafford.
It was Burnley who went ahead though and it was no surprise that the pacey Kolesoho who was the creator, racing past Gonzalo Montiel down the left and pulling the ball Amdouni to smash home a low shot from the edge of the area.
Kolesoho set up another chance for Burnley early in the second half but Charlie Taylor volleyed off target.
Forest's equaliser was worth the wait as Hudson-Odoi, making his Premier League debut for Forest, picked the ball up on the left and cut in before curling a right-foot shot into the net off the inside of the far post to give Trafford no chance.
Both sides had chances for victory but a draw was just about the right result, although Burnley's frustration was underlined by Foster's elbow into the chest of Ryan Yates which resulted in him being shown a red car after a VAR check.
"It's a moment of inexperience and something you can't excuse, but at the same time he has played his heart out for the team," Kompany said.