Lee Gregory could have put the Lions ahead, but shot over from close range.
Visting goalkeeper Jordan Archer made a fine save to deny Elliot Lee before Gregory and Chris Taylor both missed chances at the other end.
Taylor almost sealed three points for Millwall late on but his defected effort hit the post, meaning the Londoners drop to sixth in the table.
They are four points ahead of Barnsley, who have a game in hand, while Colchester remain in 23rd place, still seven points adrift of safety.
Millwall boss Neil Harris told BBC Radio London:
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"I don't think we were really quite as clinical as we have been.
"I thought we were very lacklustre in the first half and never really imposed our tempo on the game.
"I think we can focus on the fact that it's a really good clean sheet, it's another point gained.
"Having said that, I'm obviously disappointed. I thought we should have come here and won today."

The 26-year-old midfielder has signed "an initial one-year deal", with Ray McKinnon expressing delight at his first signing as manager at Tannadice.
"He is exactly the type of player we need going into this season," he said.
"Not only is he the right age and a good footballer, but he is a ball winner who is strong in the tackle and will protect our central defenders."
Murdoch had joined County last summer after his departure from Fleetwood Town.
But half of his 34 appearances came as a substitute and he followed fellow midfielder Rocco Quinn out of the Scottish Premiership club.
Murdoch, who began his career with Falkirk and had a loan spell with Northampton Town before joining Fleetwood in 2013, told United's website: "I'm delighted to be joining Dundee United - it was an easy decision to make.
"It's a massive club and I want to help them get back to the Premiership, where they should be."
Following United's relegation from the top flight, 13 players have exited Tannadice along with manager Mixu Paatelainen and his backroom staff.
While Murdoch and Quinn have exited County, manager Jim McIntyre has already recruited three defenders and a goalkeeper, Aaron McCarey from Wolves, this summer.
Christopher Routis has arrived from Bradford City, Kenny van der Weg from Breda and Erik Cikos from Slovan Bratislava.
League One strugglers Colchester United had to settle for a point against play-off chasing Millwall.
[NEXT_CONCEPT]
Stewart Murdoch has signed for Dundee United the day after his contract was terminated a year early by Ross County.