Burundians have voted for a new parliament after a night of sporadic blasts and gunshots, and weeks of violent protests against President Pierre Nkurunziza's decision to seek a third term in office.

Voting appeared slow in an election boycotted by the opposition and condemned by African and European nations as lacking the conditions to ensure it was fair. "We don't see many people," said one diplomat.

The EU, a major donor, has threatened to withhold more funds to the aid-reliant nation after Burundi ignored UN and African calls for a postponement for the parliamentary vote, and a presidential election on July 15.

Opponents say the president's bid to stand again breaks the constitution. But the government has pressed on with plans for both votes, amid the worst political crisis since an ethnically charged civil war ended in 2005.

More than 125,000 people, or over 1 per cent of the nation's 10 million population, have fled across the border, causing concern in a region with a history of ethnic conflict, particularly next-door Rwanda which suffered genocide in 1994.

Aimable Niyonkuru, 20, once a supporter of Nkurunziza's CNDD-FDD party, said he would not vote because the president had not improved the economy or delivered on other promises.

"I am really disappointed about what all politicians are doing, I think that they are not patriotic," he said.

A presidential spokesman said voting was proceeding smoothly and said African and European states could not judge the process before it was completed. He said any further delay in the votes was "out of the question".

The president cites a court ruling saying he can run for a third term and has refused to back down.

Sounds of shooting and at least two blasts were heard overnight in the capital Bujumbura, the focus of clashes between demonstrators and police.