The 56-year-old performed on some of the band's best-known albums including Rum, Sodomy And The Lash and If I Should Fall From Grace With God after joining the line-up in 1984.
Chevron, real name Philip Ryan, had performed with Irish punk act The Radiators From Space before joining The Pogues, fronted by Shane MacGowan.
He was treated for head and neck cancer in 2007 and was later given a clean bill of health, but the illness returned earlier this year and he had been told that operating on the tumour carried too great a risk.
Chevron also contributed banjo and mandolin to Pogues recordings.
Paul McLoone, lead singer of the Undertones, said: "Sincere condolences to the family and many friends of Philip Chevron. A fine musician and a great songwriter. R.I.P."