United, who went out of the Champions League on away goals to Bayern Munich on Wednesday, failed to break down a resolute Rovers side which means they remain stuck behind leaders Chelsea having played a game more.
Chelsea have 74 points from 33 games while United, whose 2-1 defeat at home to Chelsea last weekend cost them top spot, have 73 from 34 games. Arsenal are still in the hunt with 71 points from 33 games.
Liverpool's ambitions of Champions League football next season look all but over after they drew 0-0 with fellow Europa League semi-finalists Fulham at Anfield, while Manchester City beat Birmingham City 5-1 to maintain their bid for fourth place.
Roberto Mancini's City side stay in the fourth and final Champions League spot on 62 points after 33 games, four ahead of Tottenham Hotspur who have played one less.
In the day's third goalless stalemate Wolverhampton Wanderers eked out another point in their bid to retain their top-flight status, drawing at home to Stoke City.
United manager Sir Alex Ferguson made six changes to the side who started the ultimately fruitless 3-2 win at home to Bayern on Wednesday with Dimitar Berbatov and Federico Macheda up front and Ryan Giggs and Paul Scholes also included.
However, with leading scorer Wayne Rooney missing after re-injuring his ankle on Wednesday, United were toothless as for the seventh time this season they dropped vital league points in the match following a midweek Champions League fixture.
FERDINAND INJURED
"It's going to be very, very difficult to win this league now," Ferguson told the club website.
"We'll have a lifeline if Chelsea blow it, but in their eyes they have an easy game against Bolton on Tuesday. They'll expect to win that and that would put them four points clear."
Rovers' Paul Robinson was the busier of the two goalkeepers, the pick of his saves being the one that denied Antonio Valencia in the first half, but the home side looked comfortable for most of a scrappy encounter.
The visitor's problems increased late on with defender and England World Cup captain Rio Ferdinand dislocating a finger and then limping through the final minutes with a groin injury.
"We were a little unlucky with Rio's injury - that effectively reduced us to 10 men at the end," Ferguson said.
"O'Shea also got an injury and Giggs felt his hamstring in the first half. Things conspired against us a little bit in that respect but we still had enough possession to win the game."
United's neighbours City had a far more useful afternoon with an entertaining victory over ninth-placed Birmingham.
After a tepid opening, four goals in under six first-half minutes illuminated the match.
A foul on Emmanuel Adebayor allowed Carlos Tevez to give the hosts the lead from the penalty spot on 38 minutes and Nedum Onuoha quickly headed a second, which took a touch off Tevez's boot.
