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The result meant Brendan Rodgers's side remained in seventh place, six points below the European qualifying places.
Liverpool bossed the first half at Anfield but lost Stewart Downing to injury in the 25th minute and saw Mohamed Diame flash a shot over the hosts' crossbar after a dazzling dribble from wide on the right.
The home side continued to force the issue after the interval but could not find a breakthrough, with Steven Gerrard having a shot blocked on the line by James Tomkins and Luis Suarez grazing the post.
West Ham might even have snatched victory at the death, but Jack Collison's header was kneed off the line by Lucas Leiva.
Later on Sunday, Paolo Di Canio will sample Premier League management for the first time when his new club Sunderland visit Chelsea, having seen his first week in the job dominated by speculation about his political beliefs.
Di Canio, who released a statement earlier this week declaring that he is not a fascist, is bidding to reverse a slump that has seen Sunderland slip to within a point of the relegation zone.
Chelsea, meanwhile, can reclaim fourth place from Arsenal with victory.
Tottenham Hotspur will aim to tighten their grip on third place when they welcome Everton to White Hart Lane, although their opponents have Champions League aspirations of their own.
At the foot of the table, second-bottom Queens Park Rangers host third-bottom Wigan Athletic in a game that could prove pivotal to both sides' chances of escaping the drop.
In the day's other game, Europa League quarter-finalists Newcastle United tackle Fulham.
