Greg Hands represents a constituency in London, where there is strong opposition to the expansion from residents who fear the extra noise and pollution caused by many more flights.
May is demanding that all her Conservative MPs back the plan when it finally comes before the House of Commons on Monday after years of debate.
Hands, a member of the international trade ministry, said he had written to the prime minister earlier in the week to explain that he must stand by an election pledge to vote against the plan.
"As the government will be whipping the vote on Monday, this means I am resigning from the government," he said on Twitter.
A former London mayor who now represents a constituency near Heathrow, Johnson once pledged to lie in front of bulldozers to stop construction.
The opposition Labour party is also divided on the issue and is allowing its MPs a free vote, although the motion is expected to pass.
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