Branch's well-taken goal on the half-hour was cancelled out five minutes later by David Connolly's 12th goal of the campaign from the penalty spot.
West Ham, still with one eye on the automatic promotion route to the Premiership, were the brighter of the two sides early on and their neat passing game constantly caused problems to Burnley's rearguard.
It was against the run of play when the home side went ahead, Branch firing home first time with his left foot after the ball had been played inside the West Ham defence by the lively Robbie Blake.
But Burnley's joy was short-lived with Mark McGregor pulling back Bobby Zamora to gift West Ham the chance to level from the spot which Connolly accepted in expert fashion.
Zamora then saw a header smartly saved by Brian Jensen, the Burnley keeper who went on to take the man-of-the-match award after making a couple of outstanding saves to deny Nigel Reo-Coker and Marlon Harewood.
Tempers took over midway through the second half when an incident involving Hammers winger Matthew Etherington and Burnley substitute Dean West led to bookings for the visiting winger along with Branch, Blake and Jon Harley.
West had earlier been booked for a rash tackle on Etherington.
When things settled down again it was West Ham who looked the more likely winners but the best late chance fell to Burnley, Blake scuffing a shot wide after persistent work down the right from Glen Little had created a clear opening.
With West Ham pushing for promotion and Burnley far from clear of the relegation zone, a draw suited neither side but was probably just about fair.