1
general::
Finish or dispose of, especially quickly and easily. For example, We polished off the pie in no time, or If everyone helps, we can polish off this job today. This usage, dating from the early 1800s, came from boxing, where it originally meant "to defeat an opponent quickly and easily." By the 1830s it was used more generally.
American Heritage Idioms
2
general::
Phrase(s): polish something off
to eat, consume, exhaust, or complete all of something. • Who polished the cake off ? • Who polished off the cake?
McGrawhill's American Idioms And Phrasal Verbs