The Tykes twice led during the match at Oakwell, but they were unable to hold on to their advantage and have now gone four Championship games without a victory.
The South Yorkshire club have the third-worst defence in the division and Heckingbottom has criticised his side for the way they allowed Forest back into Friday’s match.
“We speak a lot about being ruthless at both ends of the pitch, and our goals were really well worked. Not taking anything away from Forest, but we gave them every single goal, and that’s what we’re disappointed with,” the 39-year-old said.
“When you go behind in this league it’s tough to come back from. We went in front twice but gave Forest a way back in [both times] and that’s how we’ve got beaten today.”
Barnsley will be hoping to return to winning ways in their next Championship fixture, when they travel to St Andrew’s to take on Birmingham City on December 3.
Motherwell held Ladbrokes Premiership leaders Celtic to a goalless draw despite playing for more than half the match with 10 men at Fir Park.
Michy Batshuayi's cheeky backheel was enough to give Borussia Dortmund a 1-0 win over Hannover 96 at Signal Iduna Park on Sunday, lifting them within a point of second-placed Schalke.