Keane defended Postecoglou on the TV show Stick to Football, criticizing the players for their role in his dismissal and suggesting they didn't make an effort to play well for him.
Keane questioned whether the players wanted to score or keep Postecoglou in his job, pointing out that even their shots on goal were not successful.
He described the players as "idiots" for not showing more commitment to the team.
Postecoglou previously led Tottenham to Europa League victory, but his time at Forest was marked by a 17th-place finish in the Premier League.
Owner Evangelos Marinakis fired him after also sacking Nuno Espirito Santo earlier this season.
Sean Dyche is now in charge of Forest, while Nuno has taken over at West Ham.
Postecoglou is currently without a job, but could potentially return to the Premier League if Wolves or Burnley struggle.