Polls opened at 06:00 local time (04:00 GMT), after a night of gunfire and explosions claimed two lives in the capital Bujumbura.
The president’s critics says the vote is unconstitutional, as he is only entitled to stand for two terms only. The government accuses the opposition of provoking violent protests.
The BBC’s Maud Jullien in Bujumbura says shooting was continuing in the streets on Monday morning.
“Bursts of gunfire could be heard in at least one district of the capital, as polling stations were about to open,” he added.
A policeman and a civilian are believed to have died.
The president’s office has described the latest protests as terrorist acts intended to disrupt the election.
Chief communications advisor Willy Nyamitwe said: “People do it to intimidate voters. They don’t want the voters to go to the polls.”
About 1,000 people are fleeing into Tanzania each day to escape the violence, according to medical charity Doctors Without Borders (MSF).
It said many families were crossing “on foot without any belongings”.
None of the president’s seven challengers has made a significant impact, correspondents say.
In addition several leading opposition parties have said they would boycott the vote.
Mr Nkurunziza’s main challenger, Agathon Rwasa, is registered as an independent candidate as his faction of the FNL party is not recognised by the government.
Critics say that a win for President Nkurunziza would be a hollow triumph that will result in him governing a bitterly-divided nation.
“The government has opted to isolate itself and go ahead with pseudo-elections,” prominent opposition figure Leonce Ngendakumana was quoted by the AFP news agency as saying after negotiations on the crisis broke down on Sunday.
Another opposition figure, Jean Minani, accused the government of being “very irresponsible”.
“They have refused to save Burundi from sliding into an abyss,” he said.
Tensions
But officials close to the president – a former guerrilla fighter, teacher and born-again Christian – argue that he is a man of the people and the only person capable of rebuilding the country after decades of civil war.
They say that he has an instinct for survival and is determined to remain in the presidential palace.
Tensions between Burundi’s ethnic Hutu majority – comprising some 85% of the 10.5 million population – and the country’s Tutsi minority have flared up regularly since independence from Belgium in 1962.
Mr Nkurunziza led a Hutu rebel group fighting the Tutsi-dominated army until a peace deal let to him becoming president in 2005.
The Constitutional Court has backed his argument that his first term in office did not count towards the two-term limit, as he was elected by MPs.
The elections are being held following weeks of protests over President Nkurunziza’s plan to seek a third term.
In May, rebel generals tried to overthrow him in a coup. After they failed, a rebellion began in the north of the country.
The African Union (AU) has not sent observers – the first time it has taken such a stance against a member state.
Some 3.8 million Burundians are eligible to vote.
By:3news.com/Ghana
Burundi votes amidst gunfire and blasts
People in Burundi are voting amid widespread tension over President Pierre Nkurunziza’s decision to stand for a third consecutive term.
