The London-based oil and gas firm said the two executives have agreed to step down from its board following an orderly transition period, expected to take approximately two months.
It said Simon Bragg was resigning as chairman with immediate effect.
Non-executive director Marcus Stanton became non-executive chairman to oversee the transition process.
Trap Oil said it wanted to reduce operating costs to a minimum while looking to maximise returns from its portfolio of North Sea assets.
The company bought Banchory-based Reach Oil & Gas for £30m in 2011 but has faced challenges following a strategy that involves farming out stakes in exploration acreage to partners who agreed to bear some of the related costs.
Trap Oil lost £10.3m before tax in 2013 when it drilled fewer wells than directors had hoped for and wrote down the value of some assets citing partner drag issues. In April Trap Oil said it had cut directors pay by 20 per cent this year and "trimmed" staff levels amid increased running costs.
Mark Groves Gidney said yesterday: "It is sad for myself and Paul Collins to be leaving the company which we both started seven years ago. However it has been a struggle for small cap explorers in the North Sea."
He said both felt the proposed strategy was in shareholders' interests.