Burton, 62, worked briefly as Albion coach under Roy Hodgson before returning to the club in May 2014, to work alongside the club's sporting and technical director Richard Garlick.
Day, 60, recruited in July 2014, was with Albion less than a year.
"It was felt changes were required," said Albion chairman Jeremy Peace.
Since appointing Garlick in January 2013 as 'sporting and technical director' to replace the FA-bound Dan Ashworth, Albion have had four head coaches - Steve Clarke, Pepe Mel, Alan Irvine and Tony Pulis.
"We are looking ahead positively to a new beginning under Tony Pulis after two gruelling seasons," Peace told the Albion club website. "All our focus now is on preparing the squad for a stronger performance.
"I am on record as saying we would be engaged in an on-going review at all levels. In Tony, we have a very experienced and hands-on operator."
Former West Ham keeper Day was head of scouting and recruitment at Championship side Brighton prior to join Albion, while Burton left his post as Arsenal's head of youth coaching to return to the Midlands.
West Bromwich Albion technical director Terry Burton and head of recruitment Mervyn Day have left The Hawthorns as part of an expected summer reshuffle.