1) to finish something, such as a meal or a bottle of liquor
2) to clean the dust off of something and make it clean and shiny again
After polishing off three 40's of 8-ball, Jamal's beer goggles caused him to mistake Rosanne for one of the Olsen twins. The next day he redefined the walk of shame.
Your wife: "After you polish off those buffalo wings, please polish off the plate."
You: "Are you kidding? You're my fucking wife. You do it."