United would have qualified with a win, but their hopes are now in the balance with the top three teams in Group B separated by only two points with one match to play.
United, who have eight points, have to visit leaders VfL Wolfsburg, who have nine, on Dec. 8 when only a win might be good enough.
PSV, who are third on seven points, will qualify if they beat bottom-placed CSKA Moscow at home in their final game.
United's Dutch manager Louis van Gaal admitted his team could have wrapped it up against the champions of the Netherlands in the first half when they dominated, but also felt his team ran out of ideas after the break.
"I think we could have finished it in the first half because in the second half we were not as good any more," he told BT Sport.
"We were afraid to play the ball but nevertheless we didn't give too many chances away.
"It is a very difficult task (now) because I think Wolfsburg is the best team in our group but it is possible.
United captain Wayne Rooney believed United were not ruthless enough. "It was a game we were capable of winning. I thought we were unlucky not to go into the break 1-0 up," he said.
"In the second half we showed a lack of composure but we are not ruthless enough. We need to score more goals and start turning these sort of games into victories."
It was United's fourth goalless draw in their last seven matches and their lack of firepower could well cost them dear this season.
The best chance of all fell to Jesse Lingard who could have wrapped up the win for United but blasted over the bar after 73 minutes.
Although both teams had their moments, overall it was a frustrating night at Old Trafford where fans paid tribute to United great George Best who died exactly 10 years ago to the day aged 59.
They held up white torchlights in the seventh minute of the match in remembrance of the No.7 shirt he wore during his 11 years at the club in the 1960s and 1970s.
(Reporting by Mike Collett; Editing by Angus MacSwan and Ian Chadband)
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