Dean Holdsworth's Sports Shield consortium and Ken Anderson's Inner Circle bought the club from Eddie Davies earlier this month.
Trevor Birch and Terry Robinson have also left the club, having helped facilitate the sale.
Bolton had a winding-up petition closed at the High Court on Monday after they paid their tax bill.
Warburton said the change of ownership and death of former chairman Phil Gartside last month prompted his decision to leave.
"It has been a privilege to serve on the Bolton Wanderers board for more than 25 years," Warburton told the club website.
"Eddie's decision to sell his shares, coupled with Phil's sad passing, seemed to mark a watershed and the right time for me to stand aside and resign as a director.
"I would like to wish both Ken Anderson and Dean Holdsworth every success as they strive to forge a successful future for this great football club."
Bolton, beaten 6-0 by Bristol City on Saturday, are bottom of the Championship and 12 points from safety with eight matches remaining.
Bolton Wanderers vice-chairman Brett Warburton has resigned following the recent takeover of the club.