France international Karim Benzema put the visitors ahead inside six minutes, before City striker Sergio Aguero levelled from the penalty spot in the 72nd minute.
Yaya Toure and Carlos Tevez came close to scoring a late winner, but Hart conceded the Premier League champions had endured a terrible campaign, with Roberto Mancini’s men without a win in Europe.
“I’m very disappointed. We had a few opportunities in this group where we could have done a lot better,” Hart told Sky Sports.
“We’re out of the competition but we’ve got a big game on Sunday [against Chelsea].”
He added: “All games have cost us [this season]. We’ve lost our two away games and not won at home.
“We’ll look at both opportunities were we lost in Amsterdam and Madrid.
“It’s been a bad campaign for us. It’s been very frustrating because we’re a very good team.
“But we’ve surprised ourselves in a very bad way in this campaign and unfortunately we weren’t able to do this business when it mattered.”