The noun onus (plural onuses) is synonymous with burden and responsibility and is interchangeable with those words. It comes from Latin, where it means load or burden, and it entered English in the 17th century via the phrase onus probandi, a bit of legal jargon meaning burden of proof. By the 19th century, onus had entered nonlegal usage.
Incidentally, onus and the adjective onerous, meaning burdensome, share common Latin roots.
Last year Illinois put the onus on shoppers to self-report their non-taxed online purchases. [Chicago Tribune]
He said the onus should be on real estate agents to self-regulate. [The Age]
This shifted the onus onto the leaders of Germany and France to forge a deal later this week. [National Post]
This formation would also put the onus on the left and right full-backs to attack more. [Guardian]