Defeat at home to Chelsea left Postecoglou winless in his first eight games in charge of Forest.
An announcement regarding the Australian's dismissal came less than 20 minutes after the final whistle.
In reaction, Hoddle said of Forest owner Evangelos Marinakos: "To leave his seat after 60 minutes that was a statement in itself. I'm surprised he has done it so quickly after the game.
"Eight games they've not gone well, but after eight matches as a manager, it's no time at all.
"Surely he has got to be given more time than that."
Meanwhile, former Chelsea winger Pat Nevin also said: "A wild shambles.
"If you are the owner and you choose the managers, you need to look at yourself sometimes.
"I do think eight games, even though it's been bad, that ain't enough to see if you are good enough."