Three people were killed and 45 wounded in clashes between protesters and police in the country's capital, Bujumbura, on Monday, bringing the death toll to 11, including soldiers and police.
As clashes continue, people have been fleeing the country into neighbouring Rwanda, out of fears of all-out fighting.
As many as 21,000 people were reported to have fled to Rwanda in April, according to UN refugee agency UNHCR.
Burundi, a small landlocked country in East Africa, came out of a 13-year civil war based on ethnic lines only nine years ago.
An estimated 300,000 people were killed in fighting between Hutus and Tutsis.
There have been recent claims and counter-claims about the distribution of threatening letters that have been circulated to people's homes.
Al Jazeera has obtained copies of the letters. They refer to people's ethnicity.
The letters - illustrated with coffins and graves - say Tutsi men, women, boys and girls will be killed if President Nkurunziza does not run for a third term.
Protesters say Nkurunziza's decision violates the Arusha Agreements that ended the civil war.
Some people blame the Imbonerakure, the youth wing of the ruling CNDD-FDD party for distributing the letters. The Imbonerakure deny that they are behind the letters.
Al Jazeera's Malcolm Webb, reporting from Bujumburu, said Burundi’s protests and the political standoff are largely not along ethnic lines.
"The ruling party was once a Hutu rebel group. It now includes a small number of Tutsis. But there are also many thousands of Hutus in the opposition," he said.
Top figure flees
On Monday, adding to the wave of people fleeing the country, Judge Sylvere Nimpagaritse, the constitutional court's vice-president, fled to Rwanda.
Nimpagaritse told AFP news agency that the court's judges had come under "enormous pressure and even death threats" from senior figures, which he refused to name, to approve the disputed candidature of the Nkurunziza.
He said that a majority of the court's seven judges believed it would be unconstitutional for Nkurunziza to stand again, but had faced "enormous pressure and even death threats" to force them to change their mind.
"In my soul and conscience I decided not to put my signature to a ruling, a decision which is clearly not lawful that would be imposed from the outside, and which has nothing legal about it," Nimpagaritse said.
Leading opposition figure Agathon Rwasa, who threatened to boycott the coming presidential election unless Nkurunziza withdrew his candidacy, condemned the country's police over the violence.
"It's a shame President Nkurunziza goes on killing innocent and unarmed people ... our police are more partisan than professional and discredit our nation," Rwasa told Al Jazeera.
The protests would lead to the "fall of dictatorship and the rise of true democracy," he said.
The violence came as John Kerry, US secretary of state, said Nkurunziza's decision to stand for a third term "flies in the face" of the constitution.
The top US diplomat said the recent unrest in Burundi was a result of public concern about that decision and "should be listened to"
Thousands of protesters joined Monday's protest with some managing to reach the centre of the capital, which they had previously failed to access amid heavy police and military presence.
The government has called the demonstrations an "insurrection" and accused protesters of stoking violence.
Our correspondent said on Monday that the protests seemed to be spreading.
"In today's protests there were more people on the streets than before, including in some suburbs in the capital that have not had any demonstrations since they began, and provinces outside the city too," Webb said.
"The police have also been escalating their response, proportional to the number of people on the streets."
According to an online crowd-sourced map depicting Burundi's protests, violence and protests have significantly increased in the past three days. Al Jazeera cannot independently verify the information in this map.