Postecoglou remains winless in his first seven games in charge at Forest, succeeding Nuno Espirito Santo.
Sutton commented on the situation in a recent podcast, stating that he's never seen a fanbase call for a manager's dismissal after just six games.
"I don't understand the purpose of singing for a manager to be sacked after six games," Sutton said.
Sutton believes that the recent success of Nottingham Forest and their affection for former manager Nuno Espirito Santo has created an "entitled" element among the fanbase.
"I think the fans are expecting too much, too quickly," Sutton added.
Sutton doesn't think the fans will change their minds about Postecoglou, and he believes the situation won't end well for the manager.
However, Sutton also thinks it's unfair to judge Postecoglou so quickly and that everyone needs time to prove themselves.