Edi Rama, the Socialist party leader, has vowed to keep up the boycott until there is “transparency of the June 28 vote.”
Rama, who is also the mayor of Tirana, claimed Berisha had stifled the judiciary, distorted the media and ruined free competition.
The square where the protests are taking place has been decked out with screens showing anti-government documentaries and signs saying "Open the ballot boxes" and "Where is my voice?".
Meanwhile, the government says the re-count would be illegal.
The Organization for Security and Co-operation in Europe, which monitored the election, said it had seen an improvement over previous polls in the erstwhile communist state, but noted certain "violations."
The protests come after foreign ministers from the European Union (EU) agreed earlier in the week in Brussels, the capital of Belgum, that Albania should be considered for official EU candidate status.