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.

Gray was a Scotland age-grade international and worked for Scottish Rugby in various coaching roles between 1994 and 2006.
He said working with the senior team is "something special indeed".
Head coach Vern Cotter added: "We're looking forward to adding his experience and passion to the group."
Scotland begin the tournament at home to England on 6 February then visit Wales and Italy before hosting France and finishing in Ireland.
Elaborating on his new role, Gray commented: "The defensive breakdown is an area highlighted by the coaches as an area for improvement, for which I've been given a very specific brief to work alongside defence coach Matt Taylor for the seven-week period of the Six Nations.
"It's a great honour and I'm looking forward to joining up with the squad in camp in St Andrews this Sunday."

Ms Rees was selected from an all-woman shortlist drawn up after the former Welsh Secretary said he would stand down at the 2015 general election.
Members of Neath Labour Party made their decision on Saturday.
The other two candidates were Karen Wilkie, deputy general secretary of the Co-operative Party, and Mabel McKeown, a former aide to Harriet Harman.

The ScotPound proposals have been put forward in a report published by the New Economics Foundation (NEF).
It argues that a purely digital SÂ£ currency, similar to Bitcoin, could boost the economy and position Scotland as a world leader in economics.
The plans follow intense debate over currency in the Scottish independence referendum.
The NEF think tank paper says Holyrood already has the powers to introduce a new currency and payment system, which could work alongside sterling.
The proposals include the digital ScotPound and a free-at-point-of-use public payment system, ScotPay, operated through an arm's length public enterprise - BancaAlba.
NEF said it wanted to drive forward the debate on money, sparked by the independence referendum, and highlight the economic and social potential of financial innovation.
It even suggests a SÂ£250 dividend for every Scottish citizen, increasing the overall purchasing power within the economy.
Lead researcher Duncan McCann added: "Scotland is in a unique position to take full advantage of current trends in digital financial innovation.
"A new currency and payment system like ScotPound would operate alongside pounds sterling, supporting small and medium businesses and putting money in the pockets of those currently excluded by the financial sector."
Mr McCann said a lot of people were moving away from cash payments and the ScotPound worked on the model of the phone becoming the new wallet.
Prof Nigel Dodd, of the London School of Economics, said: "This report is an extremely timely and welcome intervention in current debates about the way our monetary system is governed and organised, and specifically about the merits and efficacy of conventional Quantitative Easing.
"Given that the currency question played a major role in the Scottish referendum debate last year, I have no doubt that the proposals for a ScotPound will inform future discussions in Scotland. NEF's proposals are extremely clear and convincing."
Bolton Wanderers vice-chairman Brett Warburton has resigned following the recent takeover of the club.
[NEXT_CONCEPT]
Richie Gray has joined Scotland's Six Nations backroom staff as a consultant after three years as South Africa's specialist breakdown coach.
[NEXT_CONCEPT]
Bridgend councillor Christina Rees has been picked to succeed MP Peter Hain as the Labour candidate for Neath.
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Scotland could create its own digital currency to operate alongside sterling, economists have suggested.