Born a Lithuanian Jew (Meshe David Osinsky) in Kaunas province, he came alone to Britain in 1900. In 1901, he was staying in Cheetham Hill, Manchester. He started as a peddler, then set up as a general outfitter in Chesterfield in 1903 selling readymade suits bought from a wholesaler. Following his marriage to Sophie Marks in 1909 the name of the company was changed from M. Burton to Burton & Burton. They had one daughter (1910) then a son (1914). On the birth of twin boys in (1917) he gave his name as Montague Maurice Burton. However, he had not changed his name legally, which caused problems during the First World War.
By 1913 Burton had five men's tailor shops with headquarters in Sheffield and manufacturing in Leeds. He had four hundred shops, and factories and mills, by 1929, when the company went public. His firm made a quarter of the British military uniforms during World War II and a third of demobilisation clothing.
He died while speaking after a dinner in Leeds. The funeral was at the Chapeltown Synagogue.