It was the top flight's first goalless match of the season which did neither team any favours in a bid to move clear of the relegation zone.
The result left West Brom 17th in the standings on two points from five games, while 18th-placed Burnley has one from four after a dominant second-half performance.
West Brom coach Slaven Bilic was disappointed with the outcome, although he praised his side and conceded a share of the spoils was a fair result.
"I told them after the game we were looking for three points, but they should build on this," the Croatian told Sky Sports.
"We created a few chances, more than enough to score. They also had a couple to be fair. A draw was probably a fair result.
"We knew these are the games - I respect Burnley a lot - where you really need to win, or at least not lose. If we want to stay up then we have to win these types of games."
Striker Karlan Grant took centre stage for West Brom in the first half as he shot over the bar in the 18th minute before he had a goal disallowed for offside in the 37th, after home keeper Sam Johnstone had kept out an Ashley Barnes effort.
West Brom came close again in the 58th minute when visiting keeper Nick Pope parried a Grady Diangana shot and Branislav Ivanovic had the rebound blocked by a defender.
Chris Wood then twice hit the crossbar for Burnley, having failed to convert an Ashley Westwood cross before he missed with the goal at his mercy after Johnstone parried a Barnes header into his path.
West Brom missed the game's final chance as Pope produced an acrobatic save to keep out a Matheus Pereira volley from close range in the 78th minute.
Burnley manager Sean Dyche also rued his side being unable to come away with a win.
"With the quality of chances we had, we could have got three points," he said.
"It was a strange game, Barnes has one (chance), then there was a moment we deserved a penalty. I think the quality of the chances were good, but we didn't take them."
