On-loan goalkeeper Graham Stack marked his final appearance for the Lions by saving a 23rd minute Micah Hyde penalty, the highlight of a dull encounter.
Stack dived to his left to palm Hyde's penalty away after Kevin Muscat, who was also playing his final game for the home side, had pushed striker Graham Branch over in the area after 23 minutes.
Other than the penalty neither side managed a notable goalscoring opportunity in the first half with Burnley's back line well marshalled by Gary Cahill, John McGreal and Frank Sinclair, and the supporters' player of the year Darren Ward dominant for Millwall.
After the break Millwall felt aggrieved to have two penalty appeals turned down with Cahill appearing to pull back Paul Ifill on both occasions.
Millwall's best effort came from Muscat whose right-foot shot was heading for the top corner before veteran defender McGreal headed off the line after 55 minutes.
The lively Peter Sweeney hit a left-foot shot just wide of goal after 70 minutes and Mo Camara flashed a drive across goal three minutes from time but the game finished goalless.