"Every real problem we face today is the legacy of the fact she was fundamentally wrong," the former London Mayor said.
The Bradford West MP faced a barrage of twitter criticism, with one user calling him an "odious, vile, heartless opportunist."
The phrase 'no state funeral' trended on Twitter, as debate raged about how the occasion should be marked.
Controversial comedian Boyle tweeted about the "terrible news" - linking to a video of Celebrate Good Times, by Kool & The Gang.
And outspoken footballer Joey Barton said Thatcher had been "despised by many."
Many left-wingers urged restraint as the news broke.
The Daily Telegraph closed comments on its report of her death "out of respect", and political blogger Guido Fawkes shut his site "as a mark of respect".
Livingstone told Sky News the former Conservative prime minister had led millions of people out of work.
He said: "Of course she was popular, she was offering people their homes at a cut price. But she didn't build any houses.
"She created today's housing crisis, she produced the banking crisis, she created the benefits crisis. It was her government that started putting people on incapacity benefits rather than register them as unemployed because the Britain she inherited was broadly at full employment.
"She decided when she wrote off our manufacturing industry that she could live with two or three million unemployed and the legacy of that, the benefits bill that we are still struggling with today.
"In actual fact, every real problem we face today is the legacy of the fact she was fundamentally wrong."
He also said that it was to Tony Blair's "shame" that he "broadly carried on" most of her policies.
Adams, the Sinn Fein president, who was banned from the airwaves by Thatcher's government during the Troubles, said she would be "especially remembered for her shameful role during hunger strikes. Her Irish policy failed miserably."
A Facebook page, called Is Margaret Thatcher Dead Yet? had been liked by almost 40,000 people.
David Hopper, general secretary of the Durham Miners' Association, said her death was a "great day" for coal miners.
The ex-miner, who turned 70 on Monday, spent all of his working life at Wearmouth Colliery.
He said: "It looks like one of the best birthdays I have ever had.
"There's no sympathy from me for what she did to our community. She destroyed our community, our villages and our people.
"For the union this could not come soon enough and I'm pleased that I have outlived her.
"It's a great day for all the miners, I imagine we will have a counter demonstration when they have her funeral.
"Our children have got no jobs and the community is full of problems. There's no work and no money and it's very sad the legacy she has left behind.
"She absolutely hated working people and I have got very bitter memories of what she did. She turned all the nation against us and the violence that was meted out on us was terrible.
"I would say to those people who want to mourn her that they're lucky she did not treat them like she treated us."
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