United's last home victory in the Premier League came on 24 September and this was its fourth consecutive league draw on its own turf.
Mourinho's frustrations boiled over when he was ordered to leave the touchline by referee Jon Moss for kicking a water bottle in the first half after a foul was given against Paul Pogba for a dive.
West Ham made a rapid start and Diafra Sakho threatened to inflict a second home defeat of the campaign on the Red Devils with a goal after 90 seconds.
United was quickly back on level terms thanks to Zlatan Ibrahimovic but, despite introducing Wayne Rooney and Henrikh Mkhitaryan off the bench, the Red Devils were unable to find a winning goal - although Jesse Lingard did see one ruled out with 15 minutes remaining.
Another home draw leaves United in sixth, four points adrift of Tottenham, while West Ham climbed a place to 16th after a stoic defensive showing.
West Ham's last win at Old Trafford came in 2007, but Sakho's rapid opener boosted its chances of ending that run.
A clumsy foul by Lingard - one of two United changes from last week's 1-1 draw against Arsenal - gave Dimitri Payet the chance to whip a free-kick into the penalty area, where Sakho beat Ibrahimovic to place a bullet header beyond David de Gea.
Juan Mata began the United recovery with a crisp shot straight at Darren Randolph after 15 minutes, and soon after the hosts were level as Ibrahimovic atoned for his earlier mistake.
Pogba's brilliant ball over the top picked out the Swedish striker, who had drifted away from James Collins to nod past Randolph.
Pogba's next involvement led to Mourinho being ushered to the stands after the Portuguese angrily kicked a water bottle when the France international was penalised for a dive.
The Portuguese manager's mood will hardly have improved when Lingard, Ibrahimovic and Marcus Rashford were wasteful with the goal in their sights in a strong end to the half from United.
Mourinho would have been even more frustrated had West Ham caught the host cold soon after the restart, Sakho and Payet both testing De Gea's reflexes.
It was a rare attempt on either goal, though, and United looked to freshen things up with the introduction of Rooney and Mkhitaryan - who both starred in Friday's (AEDT) 4-0 Europa League battering of Feyenoord - just after the hour.
Rooney had a chance to end his 10-game run without a league goal three minutes after replacing Rashford but his effort was meek and Randolph comfortably held.
The host continued to push for a winner and Lingard thought he had snatched all three points as he converted after Mkhitaryan's effort struck the post, only for the assistant's flag to deny him.
United laid siege on the West Ham goal in the closing minutes, but the visitor's defence remained resolute and the Hammers could have won it with a late Ashley Fletcher chance that was well saved by De Gea.