The BBC's Laeila Adjovi in the capital, Ouagadougou, say the city hall and ruling party headquarters are also in flames.
Earlier, the military reportedly fired at protesters who stormed parliament.
Tear gas was also fired from a helicopter, as a crowd surged towards the presidency, reports say.
Parliament was due to consider changing the constitution so that Mr Compaore, 63, could run for office again in elections next year.
A government minister could not confirm reports that the constitutional amendment has been withdrawn, Reuters news agency reports.
Mr Compaore first took power in a coup in 1987, and has won four disputed elections since then.
The opposition has called for a campaign of civil disobedience to demand that he steps down.
"October 30 is Burkina Faso's Black Spring, like the Arab Spring," opposition activist Emile Pargui Pare told AFP news agency.
State television has gone off air after protesters stormed the building housing it and ransacked it, Reuters quotes a witness as saying.Cars were also set ablaze near parliament.
The protesters do not want Mr Compaore to change the constitution to extend his rule.
Smoke could be seen billowing from parliament.
Police had earlier fired tear gas to prevent protesters from moving in on the parliamentary building.
But about 1,500 people managed to breach the security cordon and were ransacking parliament, AFP reports.
Protesters were setting fire to documents and stealing computer equipment; cars outside the building are also on fire, it reports.
Mr Compaore is a staunch ally of the US and France, which uses Burkina Faso as a base for military operations against militant Islamists in the Sahel region.
Both France and the European Union (EU) have called on him to scrap the proposed constitutional amendment.
The EU said it could jeopardise Burkina Faso's stability. The US has also raised concern about the proposed amendment.
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