The International Trade Minister said he was standing down because Tory MPs were being denied a free vote in Monday’s debate on Heathrow.
Mr Hands becomes the latest minister to quit Theresa May’s fragile Cabinet following last year’s disastrous General Election.
The MP for Chelsea and Fulham said he intended to honour a pledge to his constituents to vote against a third runway at Heathrow.
His seat in south-west London lies in the flight path of planes arriving at and departing from Heathrow.
He tweeted: “As the Government will be whipping the vote on Monday, this means I am resigning from the Government.
“It has been an honour to serve the Prime Minister and her predecessor for the last seven years.
“I wish the PM and the Government every continuing success.”
The highest profile opponent of Heathrow expansion in the Cabinet is Foreign Secretary Boris Johnson.
In 2015, Mr Johnson pledged to “lie in front of bulldozers” to stop the expansion going ahead at the UK’s biggest airport.
The then-London mayor said: “I will lie down in front of those bulldozers and stop the construction of that third runway.”
But Mr Johnson is scheduled to be in Luxembourg for an EU meeting on Monday, so will not be attending the vote in the House of Commons.
Labour’s shadow transport secretary Andy McDonald said: “Greg Hands’ resignation piles the pressure on Boris Johnson who promised his constituents he would ‘lie down in front of the bulldozers’ to stop a third runway.
“Instead he is jetting off to Luxembourg on Monday to avoid the vote because he is too weak to stand by his promises.
“Labour opposes a third runway at Heathrow because it doesn’t meet our four tests and we are calling for a free vote for all parties on Monday.”
The Government said the proposed expansion of Heathrow would create 114,000 extra jobs in the area around the airport by 2030.
Aviation minister Baroness Sugg said: “A third runway at Heathrow would bring benefits of up to £74 billion to passengers and the wider economy, creating tens of thousands of local jobs in the process.
“This massive project will bring fantastic opportunities for UK businesses, workers and apprentices and we cannot afford to wait any longer to push ahead with it.”
A Downing Street spokesman said: “Greg Hands has served the Government with great ability and distinction over several years, and we thank him for all of his work.
“This Government is committed to expanding airport capacity at Heathrow.
“This is an important decision which will play a crucial role in securing the future of Global Britain.”