The protestors forced it off the air ahead of a planned vote on a motion to allow the veteran president to extend his 27-year rule.
Black smoke swirled in the air around parliament after demonstrators lit fires inside the building before looting computers and televisions screens and wheeling away police motobikes, a Reuters reporter said.
The crowd headed towards the presidency as a government helicopter flew overhead, shooting teargas at protesters.
Lawmakers had been due to vote on Thursday on a government plan to change the constitution to allow President Blaise Compaore - who took power in a coup in 1987 - to stand for re-election again next year, when he was due to stand down.
According to one local reporter, state television read a statement saying the proposal had been withdrawn before it was forced off air. However, the information minister, reached by telephone, said he could not confirm this.
"We did this because Blaise was trying to stay too long. We are tired of him," said Seydou Kabre, a protester in the crowd heading for the prime minister's office. "We want a change. He must go!"
Most deputies had not yet arrived for the vote when protesters, who had set up barricades outside parliament from early on Thursday, stormed the building. The crowd surged forward after police fired warning shots in the air.
A Reuters reporter saw nearby structures also on fire and vehicles outside the parliament being smashed.
Compaore's attempt to remain in power has deeply divided Burkina Faso, one of the world's poorest nations but a key player in West Africa that had positioned itself as a mediator in regional crises.
It has also drawn criticism from former colonial power France and from the United States, allies of the Burkinabe government in operations against al Qaeda-linked groups in West Africa.
Compaore has ruled the cotton and gold-producing nation with a firm grip but, in recent years, he has faced increasing criticism, including from within his own camp and the military.
"If needs be we are going to march to the presidency. We want Blaise Compaore to leave, we want change," said George Sawadogo, a 23-year-old student.
Opposition to Compaore's plan have been mounting in recent days.
Hundreds of thousands of people took to the streets of Ouagadougou and other towns across the country on Tuesday in what the opposition said was the start of a campaign of civil disobedience over the proposed constitutional reform. The government has called for restraint.
"This seems to have moved us to a situation where Compaore will have to leave power before the end of his term next year," said Gilles Yabi, an independent West Africa analyst. "It will depend on how the security forces react, but I can't imagine that Blaise will be able to finish his term if there is serious violence today."
France has called on Compaore to adhere to African Union rules preventing constitutional changes that allow leaders to stay in power. The US government has said it is concerned.
"All bets are off now," said one Western diplomat in Ouagadougou, who asked not to be identified.