Stanley, who also had goalkeeper Ross Etheridge sent off, thought they had gone ahead when Billy Kee fired in, but referee Trevor Kettle ruled it out.
Lyle Taylor had a shot easily saved by Etheridge for the Dons, and Connor Smith also hit the post.
Etheridge was dismissed for handball late on in the game.
The draw lifts the home side into the top seven on goal difference, while Stanley stay fourth.
Another famous example of referees blowing time while a goal was about to be scored was seen in the 1978 World Cup when Zico of Brazil saw his effort against Sweden disallowed after Welsh referee Clive Thomas blew the full-time whistle.
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AFC Wimbledon manager Neal Ardley speaks to BBC Radio London:
"Credit to our boys. It's a good point. We didn't create bundles, we weren't brilliant by any stretch of the imagination.
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"Accrington are a very open, free-flowing team and they've carved a lot of teams open this season.
"It's a boost for the lads to be in the play-off places but it's about where we are at the end of the season."
Accrington manager John Coleman told BBC Radio Lancashire:
"I'm delighted with the way we played but sometimes the luck doesn't go with you.
"I've never seen anything like it happen [the disallowed goal] in all my 46 years playing and managing.
"He blew as the ball was on its way into the net. You can't account for that. Everyone in the ground is flabbergasted. Referees make mistakes because they're only human and that's what makes the game of football so special because you have talking points and arguments."
Accrington Stanley were denied potential victory in the goalless draw with AFC Wimbledon when they had a goal chalked off on the half-time whistle.