Mr Greentree, who held the post at the oil and gas independent for 11 years, has been under pressure from some institutional shareholders to step down.
Lasmo has incurred deep financial problems following the pounds 1.1bn takeover of its rival, Ultramar, two years ago.
His resignation is the second high-level departure from the oil sector in less than a year. In similar circumstances Robert Horton was forced to step down as chairman and chief executive of British Petroleum last June.
Both men had staked their futures on maintaining dividends despite difficult trading conditions and cash-flow problems.
Lasmo's debts were estimated to total about pounds 900m at the end of last year, equivalent to about 90 per cent of its net assets.
The company also incurred a substantial drain on cash, which has put its 6.2p final dividend in jeopardy.
Analysts believe that the group may have to halve the final payout to bring the total cost of the dividend in line with earnings.
Mr Greentree, who earned pounds 303,000 last year, is being replaced by Joe Darby, at present responsible for the group's worldwide upstream operations.
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