Home keeper Graham Stack saved Micah Hyde's 23rd minute penalty after Graham Branch tangled with Matt Lawrence.
Millwall enjoyed the Lions share of the chances and Kevin Muscat saw an effort cleared off the line by John McGreal.
Peter Sweeney hit the post and Marvin Elliott and Muscat put efforts over after being teed up by Dennis Wise in his last game as Millwall manager.
Burnley manager Steve Cotterill:
"I was disappointed to miss out on 11th, especially as QPR lost to Cardiff," Cotterill said.
"It was a good performance and a decent day, Millwall is a very difficult place to come and get a result.
"But we could have ended up in 11th which would have looked even nicer over the summer had our penalty gone in."
Millwall: Stack, Phillips, Lawrence, Ward, Muscat (Quigley 86), Sweeney, Morris (Wise 69), Elliott, Livermore, Ifill, May (Peeters 56).
Subs Not Used: Simpson, Paul Robinson.
Booked: Sweeney, Muscat.
Burnley: Coyne, Camara, McGreal, Cahill, Sinclair, Duff, Branch, Hyde, Grant, O'Connor, Akinbiyi.
Subs Not Used: Roche, Oster, Jensen, Bowditch, Valois.
Ref: B Curson (Leicestershire).