The decision was taken because of restrictions on the media, excessive force against demonstrators and a climate of intimidation, it said.
It comes as the Catholic Church said it would no longer help organise the elections amid the political unrest.
Earlier this month, the president survived a coup attempt.
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Rights groups say at least 20 people have died in protests since Mr Nkurunziza announced on 25 April that he would seek a third term.
The UN says about 70,000 people have fled to neighbouring countries fearing political violence.
EU foreign policy chief Federica Mogherini said her team had been on the ground for more than a month.
They reported that "the election process continues to be seriously marred by restrictions on independent media, excessive use of force against demonstrators, a climate of intimidation for opposition parties and civil society and lack of confidence in the election authorities", she said in a statement.
The EU has already said it is withholding more than $2m (Â£1.3m) of funding for the elections.
Correspondents say the statement by the Catholic Church, which is hugely influential in Burundi, is another blow for Mr Nkurunziza.
Many of the country's 18 provincial electoral commissions are headed by priests, who the Catholic Church committee have asked to stand down.
Bishop Gervais Bashimiyubusa said in a statement read on Catholic radio that the church could not "endorse an election riddled with shortcomings", the AFP news agency reports.
He did not call for a boycott of the elections, but stressed that nobody should go to the polls "by threat or intimidation, or because they have been bought in one way or another".
Parliamentary elections are scheduled for 5 June and the presidential poll for 26 June.
The president's critics say his bid for a third term contravenes the constitution, which requires him to step down after two terms.
But Burundi's Constitutional Court ruled that Mr Nkurunziza's first term does not count because he was elected by parliament and not voters.
10.4m population
50 years - life expectancy for a man
2nd poorest country in the world
85% are Hutu, 14% Tutsi
300,000 died in civil war
The European Union has suspended its election monitoring mission in Burundi where President Pierre Nkurunziza is seeking a third term next month.