The question says it all: where does the phrase "beer me" come from? And, is it widely used other than in the United States?
The phrase 'beer me' meaning, "give me a beer!"
Good question, and no real origin that I can find; the oldest documented use that I could find is from The Simpsons, Episode 19, Season 2 (November 15, 1990):
Homer: "Marge, beer me! ... Don't toy with me woman."