Solskjaer's side suffered a third defeat in their last four Premier League games to put the United boss's job security back under the microscope.
Burnley's first top-flight win at Old Trafford since 1962 exposed the flaws in lacklustre United side who have made their worst start to a season since 1989-90.
The majority of Old Trafford was on its feet when chants of "stand up if you hate Glazers" echoed around the ground in reference to United's owners.
Executive vice-chairman Ed Woodward was also subjected to many other abusive chants, while those few fans who remained in the stadium booed Solskjaer and his players at full-time.
Former United defender Rio Ferdinand, a Premier League and Champions League winner with the club, slammed the performance and called for the US-based Glazer family to take action to stop the rot.
"I can't defend this," Ferdinand told BT Sport. "These young kids now in schools around the country, they are not going to be wearing Manchester United shirts.
"They are not going to be wanting to come here and support Manchester United based on what you are seeing out there. It's just not going to happen, fans are walking out after 84 minutes!
"It's an embarrassment. People at the top need to look and see this and make changes, put a plan in place that people can sit there and see where we are going now. I don't see it."
Burnley took the lead six minutes before half-time as Ben Mee jumped highest in the area to nod the ball down for Chris Wood to apply a first-time finish.
And the visitors doubled their lead in stunning fashion in the second period as Jay Rodriguez rifled a ferocious strike off the underside of the bar from just inside the area.
Fifth placed United are six points behind fourth placed Chelsea as they battle to qualify for next season's Champions League via a top four finish.
Tottenham needed a late winner from Son Heung-min to snap a four-game winless streak in the league and breathe new life into their push for a top-four spot.
The South Korean's stooping header 11 minutes from time hauled Jose Mourinho's men within six points of Chelsea, but another uninspired performance did little to suggest they are capable of chasing down his old club.
Alli ends Spurs drought
Dele Alli's opener ended a three-game run without even scoring in the league for sixth placed Spurs, but Teemu Pukki's penalty 20 minutes from time deservedly levelled for bottom-of-the-table Norwich.
Elsewhere, Ayoze Perez scored twice as third placed Leicester ended their two-match winless league run with a 4-1 victory over West Ham that was marred by an injury to their leading scorer Jamie Vardy.
Vardy limped off just before half-time at the King Power Stadium with his side 1-0 ahead thanks to Harvey Barnes' 24th-minute opener, with the winger then setting up Ricardo Pereira to fire home the second in added time.
West Ham were given a route back into the match five minutes after the interval when Wilfred Ndidi was adjudged to have bundled Sebastien Haller over in the area and Mark Noble scored from the penalty spot.
However, Leicester were given a debatable penalty of their own when Kelechi Iheanacho was deemed to have been brought down by Angelo Ogbonna and Perez made no mistake from 12 yards.
The former Newcastle player then swept home his side's final goal of the night to lift them back within three points of second-placed Manchester City.
Asked about Vardy, Leicester manager Brendan Rodgers said: "Hopefully over the next couple of days it will ease off and he will be fit.
"Jamie is a real talisman for the team but if we are going to have success it is about the team."