The Australian's future with Spurs is in doubt after exiting the Carabao Cup and FA Cup over the past week.
Hart played for Postecoglou at Celtic and told BBC Sport: "Stay strong - it's a real tough moment. You can see that he feels it, his heart is on his sleeve.
"At this moment in time he is really being tested. He is being tested in his beliefs. He is asking questions of himself - we all do in difficult moments in time.
"That's when you need support, the club behind you and the team behind you and at this moment that is what they really need."
On the importance of the Europa League, Hart said: "100%. No getting away from it now.
"That has got to be the narrative of the season. That's where the focus of Tottenham has to come to light and unite.
"It looks like they will bodge their way through the Premier League season.
"The Premier League smells Tottenham - they know how to play against Tottenham.
"The Europa League is a different animal. That's where you can create those special nights."