It's been an incredible campaign for the Hammers but their Champions League hopes have been dented by the stalemate.
David Moyes' side have recently beaten Chelsea and Liverpool while giving Manchester City a tough game.
Even after the result, though, they sit in the top four on merit, but had a tougher than expected afternoon against Sean Dyche's Clarets.
Nick Pope, who will hope to have impressed the watching Gareth Southgate, made good saves to deny Issa Diop, and Said Benrahma in either half, but the Burnley goalkeeper was rarely tested on a frustrating day for the Hammers.
It was a better afternoon for Dyche, whose side still only have one victory all season and remain in the bottom three but once again showed how difficult they can be to actually beat - a platform they must now build on to once again get themselves out of trouble.
Pope was in the wars too, getting a bloodied mouth when he took a Michail Antonio boot to the face.
And when Craig Dawson tumbled in front of Dwight McNeil the hosts feared a penalty but VAR took a close look and play continued.
Burnley threatened at the start of the second half as an early ball found Jay Rodriguez, whose close-range effort was charged down by Dawson.
Rice then became a key figure in the game creating chances with a series of runs and passes, but Burnley were able to repel the Hammers.
Burnley almost snatched a win with a swift counter attack but Matej Vydra, but he went to ground as he tried to beat Rice and referee Graham Scott immediately signalling there was no foul.
Seconds later, Rodriguez was close to turning Charlie Taylor's low ball into the net but he failed to connect, ensuring the match stayed 0-0 in Lancashire.