Sean Dyche's side went ahead from the spot in the 19th when Chris Wood picked himself up to score for the seventh time against the Hammers in the Premier League.
However, for the third time in five fixtures, the Clarets would go on to surrender their lead and finish the game pointless.
Michail Antonio scored both goals for the visitors, finishing crosses from Vladimir Coufal and Said Benrahma.
1. Nick Pope 5.8
Left completely exposed by his defence for both Antonio's goals before the break. Commanded his penalty area well, successful with a number of high claims, and saved well from Lingard deep into stoppage time.
2. Matt Lowton 4.6
Couldn't live with the pace and trickery of Benrahma, who supplied the cross for West Ham's second. Also culpable for the away side's equaliser, applying little or no pressure on Antonio and continued to be targeted as the Hammers broke forward.
3. Charlie Taylor 5.6
Deserved more engagement with his deliveries, particularly in the first half when whipping in several inviting crosses. West Ham's equaliser came down his side, though he had recovered to shut out Lingard's route to goal.
4. James Tarkowski 5.3
Not one of his tidiest displays, but had seemed to be doing enough to negate the physicality of Antonio. However, exposed when the Hammers played on the ground, between the lines and stretched the play. Missing in action for both West Ham goals.