Town failed to break down a cynical and tedious Burnley side, who had midfielder John Spicer red-carded in the first half for a two-footed lunge. The Blues came closest to a goal in the opening period when Tommy Miller hit the bar. Late on, Jordan Rhodes made his senior debut from the bench.
The Blues lined-up with Miller replacing Gavin Williams in the centre of the midfield and Alex Bruce coming in for Fabian Wilnis at the heart of the defence, but otherwise unchanged from the side which defeated Scunthorpe a week ago.
Town weren't far away from a goal in the opening minute. Danny Haynes floated in a cross from the right to an unmarked Alan Lee, whose header back across Gabor Kiraly was well-saved by the Hungarian keeper.
The Blues dominated the opening stages, but in the 11th minute the visitors broke away, Wade Elliott crossing from the right and Kyle Lafferty nodding the ball back across the Town six-yard box but with no other Claret shirt having ventured forward.
Two minutes later, Owen Garvan played a ball to Tommy Miller in the centre, 20 yards out and the former Sunderland man unleashed a left-foot strike which crashed against the bar, Jon Walters committing a foul as he went for the rebound.
Miller had another opportunity in the 16th minute from a freekick on the edge of the area, but lashed his initial shot and then the rebound into the wall. Burnley broke quickly and Andy Gray sent a ball across the Town box, but again with no one following in.
On 20 Fabian Wilnis replaced Sito, the Spaniard having suffered a concussion after an earlier blow to the head. Wilnis went to right-back with David Wright switching to the left-sided role vacated by Sito.
Robbie Blake played another low ball into the Town six-yard box in the 21st minute after a short corner, but the Blues were able to clear. At the other end, Danny Haynes shot well over after a cross had reached him at the far post.
Despite a couple of dangerous balls across the Town area, the Blues had dominated possession with Burnley clearly looking to sit back and win a point. The Portman Road crowd were unhappy with the time taken over goalkicks and throw-ins from an early stage, while the players, Pablo Couñago in particular, were unimpressed with some of the late, niggly challenges which had gone unpunished by Premier League official Phil Dowd.
The Stoke-on-Trent-based official finally started to take action against the cynical side of Burnley's game — which saw them reduced to nine-men a week ago - in the 32nd minute when James O'Connor was booked from blocking off Alan Lee as he chased a ball he had played beyond the midfielder.
Danny Haynes sent a low ball across the Burnley area in the 35th minute, but no one was there to add the final touch.
Two minutes later, Jon Walters crossed from the left, Alan Lee headed on and Pablo Couñago nodded the ball back across Kiraly, but again the Hungarian keeper pulled off a fine save.
On 41 Robbie Blake appeared to catch Alex Bruce with his studs after the ball had gone, but referee Dowd felt no foul had been committed and awarded only a throw to Burnley.
The visitors took so long to take the throw that the decision was reversed, Clarets' boss Owen Coyle subsequently receiving a talking-to from referee Dowd for preventing the Blues from having their go at restarting play.
Graham Alexander was yellow-carded for a foul on Danny Haynes with two minutes of the half remaining and was lucky not to receive an immediate second caution for sarcastically applauding Dowd's decision to book him. The referee looked as if he was going to issue another card, but in the end relented and, after a lengthy chat, waved the defender away.
A red card for a Burnley player had seemed to be only a matter of time and as the game moved into injury time, John Spicer threw himself into a two-footed challenge on Tommy Miller inside the centre circle. The Town players were clearly incensed by the tackle and the game threatened to boil over before order was restored. Referee Dowd called Spicer to him and showed him a straight red card.
While the red card, the half's final action, was correct, the referee had allowed the situation to develop by not clamping down on some of the challenges earlier in the half. Burnley had come to knock Town off their stride with a niggly, physical approach and early on had been allowed to get away with far too much.
Town ought to have been in front with Lee's early chance perhaps the best, although Kiraly had done well to save from the Irishman and also pulled off another fine stop from Pablo Couñago's header. Tommy Miller had been unlucky with his shot which struck the woodwork.
The Blues started the second period strongly, Couñago hitting a shot on the turn which deflected wide.
Town were laying siege to the Burnley penalty area, but without creating clear-cut chances. On 50 Garvan twisted and turned on the edge of the box but shot weakly to Kiraly with Lee screaming at him for the ball unmarked.
Miller shot wide and then over as the Blues continued to dominate against a Burnley side sitting back in their box with one-time loanee David Unsworth, central defensive partner Steven Caldwell and keeper Kiraly dealing with everything in the air.
Kiraly was continuing to annoy the North Stand by taking an age over each goalkick, while the Blues support had taken to counting out the seconds he held the ball in his hands, at one point reaching 16. Referee Dowd also appeared to be counting time on his fingers, but failed to invoke the now almost universally ignored six-second rule.
The Hungarian keeper saved from Walters, before Wade Elliott picked up a yellow card for a foul on Couñago. Soon after, the Spaniard volleyed wide from 25 yards.
Gavin Williams took over from the again below-par Danny Haynes, Jon Walters moving to the right with the Welshman on the left. Immediately, Williams was into the action and felt he ought to have been awarded a freekick just outside the area when he was clipped by Graham Alexander as he chased a loose ball through on goal. Referee Dowd saw no foul. If he had, the Clarets might well have lost another player.
Kiraly again saved from Couñago, then Wright's cross from the left flew beyond Lee and Walters and past the far post. The full-back was on the other end of a cross moments later, but his header looped over.
With five minutes left on the clock, Jordan Rhodes took over from Alan Lee up front. Rhodes, the son of Town keeper-coach Andy, is the first senior Blues player to have been born in the 1990s.
Rhodes almost reached a Couñago through ball a minute after coming on, but there was to be no Roy of the Rovers debut for the 17-year-old, although he won every aerial challenge, and, after a laughably inadequate four added minutes, Burnley's point was confirmed.
Town had completely dominated the second half, but never really created any clear-cut openings and Kiraly was forced into his most taxing saves in the opening period when he prevented Lee and Couñago from scoring with headers.
As so often away from home, the quality of balls into the box and the composure inside the area weren't quite good enough, although Burnley's backline, specifically Unsworth and Caldwell, defended resolutely with grit and determination throughout.
That said, it's difficult to recall a more niggly side at Portman Road than Burnley for some while and referee Dowd was far too weak in the early stages of the first half. Their red card total of three in a week seems hardly surprising on today's evidence.
Town are likely to see more sides come to Suffolk sitting back for a point in the months to come and need to find a way of breaking them down if 0-0 draws aren't going to become more common as the season progresses.
Town: Alexander, Wright, Sito (Wilnis 20), De Vos, Bruce, Garvan, Miller, Walters, Haynes (Williams 74), Lee (Rhodes 85), Couñago. Unused: Supple, Clarke. Att: 20,077.