Charlton goalkeeper Thomas Myhre proved the party pooper with an acrobatic late save that prevented the Hammers striker from celebrating his 40th birthday with a trademark goal in a game that had all the hallmarks of an end of season mid-table fixture.
Both teams made just one change as Anton Ferdinand replaced Lionel Scaloni for West Ham following their midweek defeat at Old Trafford, while Charlton brought in Dennis Rommedahl for Jerome Thomas after their victory over Newcastle United, last Sunday.
With just two places and three points separating these two sides in what could well prove to be a dress rehearsal for an FA semi-final in three weeks' time, the ninth-placed Hammers started the livelier as Dean Ashton, Marlon Harewood and Yossi Benayoun each threatened inside the opening ten minutes.
But as an increasingly lacklustre first-half unfolded, the dominant East Enders continued to prod and probe the south Londoners' defence without seriously troubling Myhre.
Apart from a Bryan Hughes scuff and Radostin Kishishev's speculative, deceptively-bouncing 25-yarder, Charlton left Shaka Hislop a redundant spectator.
As the interval neared, however, his counterpart Myhre was relieved to see Ferdinand's angled shot rip into the side netting, before the Norwegian 'keeper did well to save Harewood's low half-volley at the base of his right-hand post to keep it goalless at the break.
The biggest cheer of the afternoon came as Sheringham emerged for the restart in place of the anonymous Ashton.
And how the locals were willing the 51-times capped former England international to mark his 40th with a goal.
But a string of early Charlton corners suggested that West Ham were not going to boss the second half, too.
Indeed, Darren Bent soon forced Hislop into his first meaningful save of the afternoon before Luke Young - who had earlier been booked for scything through Hayden Mullins - retired with a twisted ankle sustained in a challenge with former team-mate Paul Konchesky.
Hughes, Marcus Bent and Matt Holland also tried their luck without success, while at the other end James Collins sent a rising 30-yarder inches over, before Benayoun drilled just wide of the far post.
More substitutions followed as Thomas entered the fray but it was West Ham's new arrival - Bobby Zamora - who almost made the breakthrough when he met Matthew Etherington's 78th-minute corner with a bullet-header that was cleared off the line by Kishishev.
And when Etherington sent over another late corner, Myhre denied Sheringham the perfect present when he brilliantly clawed away the veteran's far-post volley to secure a point for his side.
That left the 34,753 crowd dispersing into the early evening East End air, well and truly hoping that if Charlton do overcome Middlesbrough in their quarter-final replay, any FA Cup semi-final will be a far more watchable affair than this tepid Premiership derby.