A long time ago I had some involvement with Bass one of the big Burton breweries and at the time the largest pub landlord in Britain. The story I heard there was that the phrase originated during the first world war in the trenches as a euphemism for someone having died.
It was considered bad luck to talk about death and so like the similar 'buying the farm' or 'he's bought it' as a way of saying he's gone to a better place (e.g. in this case down the pub), the phrase entered into popular parlance.
There was also an advertising campaign between the wars themed on someone missing from a group with the title 'he's gone for a Burton' most probably building on this cultural meme. So there is evidence that the phrase goes back considerably earlier than WW2.