Burnley led at half-time after Bashir Humphreys' cross deflected off United defender Ayden Heaven and found the far corner, but the visitors looked set for victory when Slovenian striker Benjamin Sesko struck twice inside 10 minutes early in the second half.
The Slovenian forward fired in from Bruno Fernandes’ incisive pass, then showed fine technique to steer Patrick Dorgu's cross past Martin Dubravka as United took a deserved 2-1 lead.
But Burnley hit back once more as Jaidon Anthony was given time and space to turn before firing brilliantly into the top corner to deny the visitors all three points.
Interim boss Darren Fletcher, who scrapped Amorim's wing-back system, saw his side create plenty without finding a winner. Matheus Cunha and Patrick Dorgu had efforts cleared off the line, Lisandro Martinez saw a goal disallowed for a debatable push, Sesko missed chances for a hat-trick, and substitute Shea Lacey hit the bar.
United fans voiced their anger at the club's leadership before kick-off, targeting Sir Jim Ratcliffe and Joel Glazer, while also singing in support of Ole Gunnar Solskjaer and Michael Carrick as potential caretaker options.
"The fact we have drawn is a massive disappointment. If you look at the context of the game - the chances we created, goalline clearances, a disallowed goal, 30 shots, lots of attacking play - we created more than enough to win," said Fletcher.
On the disallowed goal, he added: "It just looked like normal things that happen at every corner so it's surprising that decision was given. If you look at the context of the Premier League this season and the physicality that goes into set-pieces, I have seen way worse on almost every set-piece, so I am massively surprised the referee has got involved there. It's something we have to accept but there was a lot of frustration."
Fletcher was full of praise for Sesko's two-goal contribution on the night.
"I was so happy for him. I spoke to him yesterday and we presented a video showing his movement and showing his goals - just showing how it is coming and he needs to keep believing and the players need to get more service to him. He did that tonight and scored two great goals. A real positive night for Ben."
Arne Slot repeatedly turned to his own words to explain Liverpool's frustration and concern after Conor Bradley was injured late on in the 0-0 Premier League draw with Arsenal on Thursday night.
Mikel Arteta said his Arsenal side deserved major credit for a "remarkable" festive run after a goalless draw with Liverpool that still left the Gunners six points clear at the top of the Premier League.
Mikel Arteta has said Arsenal "have a point to prove" as they prepare to host defending Premier League champions Liverpool at the Emirates Stadium tonight.
Thomas Frank has said Tottenham's latest setback was "extremely painful" after a dramatic stoppage-time defeat at Bournemouth, as the Spurs head coach faces fresh scrutiny following a chaotic night on and off the pitch.
