Where does the phrase gone for a Burton come from?
Gone for a Burton comes from "Going for a Beer". During the Second World War it was used in black humour when someone died in battle. Usually the RAF.
They had not died or had gone missing, they had only slipped out for a Beer. Burton's being one of the biggest Breweries and Brands of Beer at that time.
This response is incomplete.
The phrase actually appeared in print, for the first time, on the 30 August 1941 in the New Statesman. It was first used when referring to the death in action of pilots in WWII.
From where did the expression spring? Who knows. Was it derived from a Spanish Burton (a kind of complex pulley arrangement for Royal Navy ship stowage - see http://compass.seacadets.org/pdf/nrtc/sn/14067_ch4.pdf and see p.34); used figuratively as fitted for a suit from the tailors Montagu Burton - eg. as one would say 'he's being fitted for a wooden overcoat' i.e. a coffin; beer as produced by a brewery located in Burton-on-Trent (this area is actually a well known brewing town in the Midlands); is it rhyming slang based on the name of that town? 'Went' - 'he's at Burton-on-Trent [went]'; based on the name of the town, going for a beer , it could be said 'he's gone for a Burton'. It's interesting to note that the brewing company Burton Brewery Co Ltd closed its doors in 1935. It was a minor league player in the beer stakes before it 'went for a burton' so was not missed on its closing down.
Which one to choose? Well take you pick. It could be any one of them or others. But knowing the frailties of human nature, my bet is on the phrase having a connection to beer.
Addition to answer
The Burton Brewery Co Ltd may not have existed after 1935 but the brewing industry consisted then of many more breweries.
Still today (2010) brewing is a major part of Burton upon Trent's industry with two major breweries and several micro breweries still operating. So in effect you can still "Go for a Burton" in Burton.