United, who are in eighth place eight points behind leaders Manchester City and second-placed Arsenal, lost further ground after missing a host of chances as their former goalkeeper Tom Heaton produced a superb performance at Old Trafford.
The 30-year old, who never got a look-in at United during his 2005-2010 stint at the club, made 11 saves as the home side enjoyed 72 percent of the possession, with 37 shots on goal.
With Heaton producing stop after stop in the first half, Mourinho's frustrations grew, and he was sent to the stands by referee Mark Clattenburg at the interval after the Portuguese protested over a first-half penalty claim being waved away.
Charged with misconduct in the wake of his comments about referee Anthony Taylor before United's 0-0 draw at fierce rivals Liverpool last week, Mourinho will now face further disciplinary action after falling out with Clattenburg.
"I don't know what happened," Mourinho's assistant and compatriot Rui Faria told the BBC.
"It is not important what Jose said. What is important is what the referee writes in his report.
"We dominated from the first second. We tried everything to score. With 10 men we kept that same mentality. The Burnley keeper was the man of the match."
The United manager complained to Clattenburg at halftime after right back Matteo Darmian had gone down in the box during the first half following minimal contact with a Burnley defender.
When he reappeared at the start of the second half, he headed for the front row of the stands before moving to the VIP box.
The home side continued to lay siege to Burnley's goal and it took a breathtaking save from the inspired Heaton to keep out an acrobatic overhead kick by Ibrahimovic, who has now scored just one goal in his last nine games for United in all competitions.
Ibrahimovic and the lively Juan Mata, who also had several shots parried by Heaton, hit the woodwork within a minute before Herrera was shown a second yellow card for a clumsy challenge in the 68th minute.
Despite their numerical disadvantage, United continued to press forward relentlessly, Wayne Rooney and left back Luke Shaw both going close before Ibrahimovic sliced a shot wide at the far post with the goal gaping.
(Writing by Zoran Milosavljevic; Editing by Neville Dalton)
