The Hammers missed out on three points on a weekend when fourth-placed Tottenham lost, meaning they failed to capitalise on a chance to close the gap.
Tottenham remain fourth on 57 points from 32 games, ahead of Manchester United (54 from 32), Arsenal (54 from 31) followed by West Ham on 52 points from 33 games.
“I think that was a game where we missed an opportunity,” Moyes told reporters. “Today we had an opportunity to challenge the teams closer to the top and we’ve given that up today by the result.”
Wout Weghorst had nodded Burnley ahead in the 33rd minute, while Maxwel Cornet missed a penalty on half-time. Tomas Soucek equalised for West Ham from a 74th-minute free-kick.
There was a lengthy first-half delay prior to Weghorst’s goal when Burnley’s Ashley Westwood went down with a serious injury under a tackle from Nikola Vlasic, with play halted for eight minutes. Vlasic fell to his knees in shock as he reacted to the nasty injury.
Burnley interim boss Mike Jackson said: “He’s gone to hospital and we’re waiting for an update. We’re hoping he’s OK - our thoughts are with him. You could see it wasn’t good from the reaction of the players. He looked distraught, but fingers crossed.
“When you see Declan Rice come over and say it’s not good, you know it’s not. The lad himself put his hand up, and he doesn’t usually go down.”
Elsewhere, Bruno Guimaraes headed in a late winner to complete his own double and round out a 2-1 win for Newcastle United over Leicester City.