The club announced that the 61-year-old's contract, which was reported to end this summer, would not be extended.
Chairman Steve Gibson said: "I'd like to thank Tony for his hard work and commitment over the past 18 months. He has been a joy to work with and a complete gentleman.
"Two years ago we invested heavily in the squad in terms of transfer fees and salaries in an attempt to regain our Premier League status. This happened before Tony's arrival.
"Over the past 18 months Tony has been proactive, not only in the ambition to win football matches and promotion to the Premier League, but also in addressing our financial position in order to comply and respect the Championship fair play criteria.
"He has ensured that the club is stable and strong for the future. He leaves us in excellent condition."