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."
League One strugglers Colchester United had to settle for a point against play-off chasing Millwall.