South African Arthur, who claims that he was relieved of his duties 16 days before the first Ashes Test because "he did not understand the Australian way", hopes to receive $4m (£2.4m) in compensation.
However, Warne believes that the 45-year-old is acting unreasonably.
He told Sky Sports News: "I think Arthur should be very grateful for the opportunity [to coach Australia] to start with.
"He did the best he could do and put his heart and soul into it, so it sounds a bit of sour grapes to me. If that's your gripe go behind closed doors, but I think he should sit back and be quiet as $4m is out of order.
"It is very disappointing that he wants money and perhaps he should go back to homework gate and start writing some lines and send some papers to Cricket Australia."
Arthur hit the headlines earlier this year when he axed four players for a Test match in India because they failed to complete a team assignment.