I'm sure Ewood's link-site is correct about the connection with Burton beer, but - just as a matter of interest - if you click http://www.word-detective.com/041899.html#burton
you'll find another couple of theories. They're a lot less convincing, but perhaps worth knowing about at least.
As I remember, during WWII, the expression ''Went for a Burton'' referred to someone in the RAF who had been killed. The tailoring chain of Burton was among manufacturers who made 'demob suits' - civilian suits issued to people leaving the armed forces. If killed, a person would be considered (in a wry way) by his colleagues to have left the service, i.e. picked up his demob suit.