Middlesbrough failed to have a shot in the opening 45 minutes, as the home side dominated play.
However, a series of saves from Jason Steele ensured that the sides went into the break at 0-0.
Burnley made a strong start to the second half, but Middlesbrough started to play their way into the match, with substitute Mustapha Carayol looking dangerous for the visitors.
Some open play in the final five minutes saw both sides have chances to score a winner, but both defences held strong to ensure that the tie ended without a goal.
The draw keeps Middlesbrough in the playoff places, while Burnley remain 11th.