Click http://www.word-detective.com/041899.html#burton for a website which offers several possible explanations. It certainly thinks Tartanwizard's is most likely, but it does offer other possibilities. Certainly, its first appearance in print was in the 'New Statesman' magazine of August 1941. The Oxford English Dictionary, incidentally, says: "None of the colourful versions of the origin of the phrase is authenticated", so perhaps none of the suggestions we've offered is correct!

I always thought it came from an advertising campaign run Burton Ales with the tag line "Gone for a Burton.." but not sure

I think Moog is on the right lines. I heard that the ad campaign ran pictures of famous sets of people with one missing ( i.e. The Beatles without John Lennon ) and the tag line ' He's gone for a Burton '.