The phrase "onerous responsibility" means having a difficult or burdensome task to complete or a duty that requires a lot of effort and can be challenging to fulfill. Full definition
«Calling people to exercise their democratic rights in a situation where their security can not be guaranteed is a most onerous responsibility.» (bbc.com)
«That this attack is as a result of the failure of security agencies particularly the JTF, the Navy and the police whose onerous responsibility is to provide security and protect lives and property of the citizenry,» the monarch said. (dailypost.ng)
Yet for shouldering this onerous responsibility, Mr Greenspan is paid a comparatively modest $ 140,000 a year. (economist.com)