The Clarets were unable to arrest their winless home form, which now stretches to 14 league games, as the Canaries settled for a point, which brought an end to their run of 16 successive defeats in the top flight.
Sean Dyche's side, who have now failed to beat the last six promoted sides they have faced on home turf, enjoyed the lion's share of possession and territory during a drab affair, but failed to test goalkeeper Tim Krul.
Referee Kevin Friend dished out the yellow cards and waved away numerous penalty appeals in an untidy first half, which saw the visitors go closest to scoring when Max Aarons pulled an effort wide of the far post.
Daniel Farke's side also went close after the interval. Jay Rodriguez unlawfully halted Ozan Kabak's run through the middle and Nick Pope, who missed out on selection for Gareth Southgate's England squad, had to be at his best to divert Mathias Normann's set-piece to safety.
Krul, an ex-international team-mate of Erik Pieters, beat away baptised the second half with a save from McNeil, who would later squander a golden opportunity to wrap up the points when failing to convert substitute Johann Berg Gudmundsson's cross.
Rodriguez, who had replaced Matej Vydra before the break, failed to connect with the goal at his mercy once Krul had attempted to cut out a cross before defender Matt Lowton powered a header over from Josh Brownhill's delivery.
Here are the ratings.
Referee Kevin Friend dished out the yellow cards and waved away numerous penalty appeals in an untidy first half, which saw the visitors go closest to scoring when Max Aarons pulled an effort wide of the far post.
5. Nathan Collins 7
Made a very encouraging first start in the Premier League. Telegraphed passes to nip in front of his marker and win the ball, good in the air, positionally astute, slowed Norwich's counters and timed his tackles to perfection, which denied Sargant after the break.
6. Aaron Lennon 5.5
Involved in fits and starts. Had the beating of Dimitrios Giannoulis early on, when going around the outside, but execution wasn't up to scratch. Played with a smile, saw a first half shot blocked, but faded and replaced by Gudmundsson on the hour.
7. Dwight McNeil 5
Could be accused of being a little bit selfish at times, taking far too long to release the ball, and sometimes took the wrong option. Offered glimpses of what he can do when wriggling free of challenges and drifting infield, but missed a gilt-edged chance late on.
8. Ashley Westwood 5.5
Hit and miss. His composure on the ball allowed the Clarets to recalibrate in rare moments when the visitors had threatened. Spread the ball well, but failed to test Hanley, Gibson or Kabak when getting the ball forward quickly. Conceded possession regularly.