Both sides had the opportunity to claim all three points, but it was the goalkeepers who came out on top in the first goalless draw of the season in the top flight.
Ashley Barnes brought an excellent save from Sam Johnstone in the first half while Nick Pope did exceptionally well to keep out Grady Diangana, Karlan Grant and Branislav Ivanovic in quick succession after the break.
Chris Wood should have put the away side ahead afterwards when clipping the bar from Ashley Westwood's delivery before Pope preserved his clean sheet when turning Matheus Pereira's attempt around the post.
The Clarets had one last chance to win it late on, but Johnstone came out on top of his battle with Barnes once again when beating the striker's header away.
Chris Wood should have put the away side ahead afterwards when clipping the bar from Ashley Westwood's delivery before Pope preserved his clean sheet when turning Matheus Pereira's attempt around the post.