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.
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