Police shot at the demonstrators in a bid to disperse them, with witnesses reporting several injured in the capital Bujumbura. An AFP journalist saw three people with bullet wounds.
The fresh unrest in the small central African nation came after the end of a two-day truce that followed a week of violent political protests over Nkurunziza's attempt at securing a third term in office.
A few hundred protestors gathered in a suburb of the capital Bujumbura, shouting at police, who have for days blocked roads to prevent demonstrators from moving into the centre of the city.
One group however broke through and reached the centre, the first time they have managed to do so since protests began. But once there they were quickly dispersed by police, who hurled stun grenades and fired tear gas.
Nkurunziza has been in power since 2005. His supporters, however, say he is eligible to run again, since his first term in office followed his election by parliament - not directly by the people as the constitution specifies.
The opposition protesters announced a truce over the weekend but returned to the streets on Monday demanding again that Nkurunziza backed down.
"Let us through!" demonstrators shouted, as soldiers sought to ease tensions between the police and demonstrators.
Since the protests started, the army has regularly come between the police and demonstrators to avoid further clashes and the protesters believe the soldiers are neutral.
At least 10 people have died and scores more have been hurt since the protests began last weekend. Nearly 600 people have also been arrested, according to police.
"We have two camps fixed in their positions - and no one is willing to back down," said a diplomat.
The government linked a grenade attack that killed three people, including to police officers, in the early hours of Saturday to the opposition protests and branded the demonstrators "enemies of the state".
The issue of the third term has both supporters and opponents within the security services.
"There are some pretty serious differences within the security forces," added the diplomat.
On Sunday, the army's chief of staff pledged the military's loyalty to the country's authorities after the defence minister had declared the army's neutrality.
General Prime Niyongabo said the military "remains and will remain a republican and loyalist army that is respectful of the laws and rules of Burundi and of those who govern it."
A statement by Defence Minister General Pontien Gaciyubwenge on Saturday declaring the army's neutrality and calling for an end to attacks on citizens' rights appeared to flag up possible divisions in the army.
The unrest erupted after the ruling CNDD-FDD party designated Nkurunziza, a former rebel leader and born-again Christian from the Hutu majority, as its candidate in presidential elections due to be held on June 26.