Thompson had another rush of blood on the half-hour mark and was dismissed following another rash challenge on Mills.
Burnley manager Owen Coyle replaced the injured Wade Elliott with Alan Mahon as he sought to reorganise his side.
Opposition manager Sean O'Driscoll took the bull by the horns by replacing defender Sam Hird with striker James Hayter.
The best chance of the opening half came in stoppage time when Burnley goalkeeper Brian Jensen raced out of his area but could only head the ball straight to Jason Price who, with the goal at his mercy, placed his first-time shot wide of the target.
Burnley had an opportunity just five minutes into the second half when a Robbie Blake free-kick was met by the head of Chris McCann but he placed his effort straight at Doncaster goalkeeper Neil Sullivan.
Gradually Rovers began to get on top and Jensen was forced to make saves from Martin Woods, Brian Stock and Price.
The visitors' patient build-up saw a further chance for substitute Paul Heffernan but he blazed over.
At the other end in a rare Burnley attack, Chris Eagles dragged a shot across the face of goal.
In the end, Burnley were indebted to goalkeeper Jensen who, in stoppage time, made a superb double save from substitute Hayter which prevented the visitors from claiming all three points.