Tommy Miller hit the bar and Alan Lee and Pablo Counago were both denied by Gabor Kiraly saves as Blues were left frustrated by the 10 men of Burnley.
Lee's header was pushed out by Kiraly inside the first two minutes, Miller smashed a shot against the woodwork, with Counago then denied with a close range header as Town created the clearer openings in the first-half.
Burnley were reduced to 10 men just before the break when John Spicer was red carded for a two footed challenge on Jon Walters and the Clarets were forced to defend inside their own box for all the second-half but hung on for the draw.
Town made two changes from the team that beat Scunthorpe last week, Alex Bruce and Miller coming in for Gavin Williams and Fabian Wilnis.
Burnley included former Blues' loanee David Unsworth in their line-up, another ex-Town loanee, Alan Mahon, missing out through injury.
Blues should have been ahead inside two minutes, Counago and Danny Haynes linking up well and Lee left clear six yards out but his header was pushed away by Gabor Kiraly.
Kyle Lafferty headed across goal at the other end after Wade Elliott's cross before Miller rattled the bar with a terrific strike from 20 yards.
The midfielder then saw his free-kick blocked by the Burnley wall after Counago had been fouled by Unsworth, the Clarets breaking at pace on the counter with Andy Gray's cross escaping Lafferty at the far post.
The Clarets conjured up another opening from a corner but Robbie Blake's cross was cleared by Wilnis, who was on for Sito, the defender taken to hospital with mild concussion.
Walters, Town's most effective front player in the first-half, cut inside after a 50 yard run but saw his shot blocked by Steven Caldwell.
Counago and Walters then linked up to find Haynes at the far post but he shot high over the bar.
James O'Connor was yellow carded on the half hour for blocking off Lee, with Blues almost ahead soon after - Counago's header set for the net until Kiraly clawed the ball away.
Lee and Burnley boss Owen Coyle clashed on the touchline as the game grew in intensity, Graham Alexander booked for dissent, with Bruce and Blake tangling as well.
The half finished with a red card shown to Spicer by referee Dowd for a reckless challenge on Walters that prompted a melee in the centre circle that quickly fizzled out.
Garvan had the first opening of the second-half, weaving inside Caldwell but his right foot shot lacked power. Miller then shooting wide after Haynes' cross had felt to him 25 yards out.
Haynes' trickery down the right flank carved out another opening but his cross was hacked away by O'Connor.
Counago, who after a slow start was beginning to have a major say in the game, fed Walters on the edge of the box, Miller firing over from the knock-back.
Burnley made their first change on the hour, Steve Jones replacing Blake.
Kiraly saved from Walters as Town camped around the visitors' goal, Burnley then picking up a third booking with Elliott cautioned for a foul on Counago. Ade Akinbiyi replaced Gray for the last 20 minutes. Williams coming on for Haynes in Town's second change.
Lafferty's perseverance created the Clarets' first opening of the second-half on 75 minutes, Bruce clearing at the near post. Akinbiyi then firing high and wide.
Counago forced Kiraly into a low save as the Spaniard twisted and turned inside the Burnley box looking for the breakthrough.
David Wright sent a header over the bar before Town made a final change, Jordan Rhodes coming on for his debut five minutes from the end, Lee making way.
Bruce fired over with Town's last effort as Burnley held on, the draw stretching Blues' unbeaten run at Portman Road to 16 games.
Town: Alexander, Wright, De Vos, Bruce, Sito (Wilnis 20), Haynes (Williams 73), Garvan, Miller, Walters, Lee (Rhodes 85), Counago. Subs not used: Supple, Clarke.
Burnley: Kiraly, Alexander, Caldwell, Unsworth, Harley, Elliott, J. O'Connor, Spicer, Lafferty, Blake (Jones 59), Gray (Akinbiyi 70).Subs not used: Jensen, G. O'Connor, Jordan. Sent-off: Spicer. Booked: J.O'Connor, Alexander.
Att: 20,077
