Several polling centres were attacked during the night in the capital, Bujumbura, and in some provinces ahead of the vote, but the attackers failed to damage electoral materials, police said Monday morning.
Some 3.8 million Burundians are eligible to vote in the polls, which the opposition and civil society groups are boycotting, claiming they will not be free and fair.
While the parliamentary and local elections have been set for Monday, the presidential vote is not scheduled to take place until July 15.
Burundian authorities have refused to delay the elections despite repeated calls from the international community.
UN Secretary General Ban Ki-moon has called for the poll to be scrapped, saying a fair vote would be impossible due to the tens of thousands of people that have fled to neighboring countries for fear of violence.
More than 70 people have been killed since Nkurunziza announced in May that he would seek a new term in office. The unrest triggered a brief military coup, but the president was soon reinstated.
On Sunday, the speaker of Burundi's parliament told FRANCE 24 that he had fled the restive Central African nation to Belgium, denouncing Nkurunziza’s “illegal” bid to seek a third term.
“I was forced to quit the country due to the unrest caused by the president’s insistence to seek a third term bid which is illegal, which is unconstitutional,” National Assembly Speaker Pie Ntavyohanyuma said in an interview in Brussels.
"I would like to say to him that the mandate he wants to have is illegal. I would like to say to him that forcing through the election is senseless."
According to Ntavyohanyuma, the country’s mediators have all urged the president to postpone the elections, amid the worst crisis the country has faced since the civil war ended nine years ago.