Victories for top two Southampton and Reading meant there was little comfort for the Hammers to take after Kevin Nolan and James Tomkins secured a fifth successive draw on the back of a dismal first-half showing.
Martin Paterson sent Burnley in two goals to the good at the interval after Marvin Bartley opened the scoring with a fine individual effort.
Carlton Cole and Martin Taylor both spurned good chances for West Ham and their challenge appeared to have petered out until the second-half rally arrived.
The result left Allardyce's men four points shy of Reading in second with a game in hand, while opposite number Eddie Howe has not tasted victory for seven games.
The first clear-cut chance fell to the home side when recalled left-back Ben Mee found space to cross for Paterson, who sent his free header wide.
In the ninth minute West Ham should have taken the lead. Cole latched on to Taylor's through-ball and rounded Clarets goalkeeper Lee Grant, but scuffed his shot into the side netting with the goal gaping.
Cole then fired high and wide after Joey O'Brien's cross deflected into his path and he would soon play for his profligacy.
There appeared to be little danger when Paterson and Ross Wallace combined to tee up Bartley midway inside the West Ham half. The midfielder proceeded to charge past Danny Collins and unleash a low drive that Robert Green could only touch into the corner of his net.
The Hammers thought they had an immediate response when Nolan planted home a header via the underside of Grant's crossbar, but his effort was ruled out for a push on David Edgar.
Allardyce's frustration was compounded as Burnley doubled their lead 10 minutes before the half-time.
West Ham failed to heed their earlier warning as Mee was once more allowed to advance down the left, with Paterson converting successfully at the near post on this occasion.
Clarets defender Michael Duff almost handed the visitors a lifeline in first-half stoppage-time when he missed a back header and Taylor fired straight at Grant.
Taylor and Cole were punished for their missed chances as Allardyce sent Sam Baldock and Nicky Maynard at half-time and reverted to 4-4-2.
Paterson outmuscled Tomkins as Burnley looked to increase their advantage in the opening minutes of the second half.
Baldock had a 25-yard effort deflected wide, but West Ham threatened only sporadically until their comeback began in the 68th minute.
O'Neil lofted a speculative ball forward and Nolan capitalised on a moment's hesitation from Duff to lob the on-rushing Grant.
Two minutes later it was all square when Tomkins found the decisive touch after Maynard flicked on Mark Noble's free-kick, awarded when Wallace felled George McCartney 30 yards from goal on the left.
West Ham almost had a third goal in five minutes as Baldock sprung the Burnley offside trap and fired a low cross beyond Grant that Kieran Trippier hacked clear with Baldock and Nolan in close attendance.
Baldock spurned a golden opportunity to take the points at the death, but blazed over from six yards when substitute John Carew chested into his path.