In the first half, Alex Palmer denied his former Baggies team-mate Jeremy Sarmiento when clean through.
Jaidon Anthony also headed in a 55th-minute left-wing cross from Lucas Pires but it was harshly ruled out for a perceived push in the back on Callum Styles.
Pires then hit the left upright three minutes later with a low shot - but, despite the determined efforts of both sides, it was a largely forgettable encounter.
After shut-outs at both ends in their previous two home games against Millwall and Cardiff City, the fifth successive goalless half of football at The Hawthorns contained very few moments to worry the scoreboard operator.
Aside from an eighth-minute header by Zian Flemming that was blocked, Burnley had the only other on-target effort of the first half when former Baggies loan man Sarmiento had a great chance after being put clear, only for home keeper Palmer to save with his legs.
The Baggies worked busily, forcing a couple of frantic Burnley scrambles but the closest they came was Styles' fierce shot which went just over.
It was much the same story after the break. Albion had one effort blocked, their only shot on target, while Burnley had their two close shaves - but it still all adds up to just three goals scored at either end in the Baggies' first seven home games this season.