Up to 30,000 protesters, who accuse the government of rigging Sunday's vote, demonstrated against the result in Chisinau, the capital, on Tuesday.
Between 100 and 200 people stormed parliament and Voronin's presidential office, looting and setting fire to the buildings.
The demonstrators say that voter lists were full of names of people who had passed away and that many of the lists included people who had moved elsewhere or were working abroad and yet had still voted in their original constituencies.
Members of the press in Moldova have also spoken to Al Jazeera of intimidation by the police, with several journalists having been detained over the last few days.
Aneta Grosu, editor-in-chief of Ziarul de Garda, an investigative newspaper, said: "We are collecting witness reports about the fraud. It is already clear that the election was stolen.
"Two of our reporters were stopped and intimidated in the street by men in civilian clothes and their tapes removed. Thanks God none was beaten - just kicked a few times."
The allegations have been dismissed by the government and the Communist party who say that the opposition cannot accept the democratic result of the election.
Voronin has accused neighbouring Romania of being behind the protests, ordering the Romanian ambassador to leave the country.
The European Union has since urged Moldova to resume normal relations with its neighbour.
In a sign of further unrest, about 200 anti-communist protesters gathered in Chisinau on Friday, calling for the resignation of the government.
At a rally in front of the government building, demonstrators chanted "down with
the Communists!'' and "resignation!"
Alina, a student gathered with the protesters, said she doubted the recount would go ahead.
"He ordered the recount, but he can see that there aren't many people here and may in the end refuse to do it," she said.
"We are not at all sure he will do it. We want freedom and democracy."
On Thursday, international press freedom groups criticized a decision by Moldova to ban 18 journalists working for Romanian and international media from covering the unrest.
Reporters Without Borders and the Committee to Protect Journalists condemned what they called an "unfair and dangerous decision by the Moldovan authorities".
Paris-based Reporters Without Borders said the ban prevented proper media coverage of the situation.
The journalists were stopped from entering Moldova on Wednesday.
Other Romanian journalists were sent back from Chisinau airport.
Voronin, who has been in power since 2001, cannot seek a third consecutive term under constitutional limits.