The government order comes after President Pierre Nkurunziza launched a campaign "to moralise society".
A government spokesman said a crackdown on informal relationships was needed to combat a population explosion.
He said too many schoolgirls were getting pregnant and men were taking advantage of women by cohabiting with several simultaneously.
Burundi has been in crisis since 2015 when Mr Nkurunziza, a born-again Christian, announced he would run for a controversial third term.
Interior ministry spokesman Terence Ntahiraja told AFP news agency that church and state-sanctioned weddings were the solution to the country's population explosion - and a patriotic duty.
"We want Burundians to understand that everyone is responsible for his life, we want order in this country," he said.
"All this is done within the framework of the patriotic training programme," he said, referring to an initiative launched by President Nkurunziza.
It is not clear exactly what sanctions those not tying the knot will face.
However, one farmer quoted by AFP said local officials had already threatened him and his partner with a fine and said any child born out of wedlock would not qualify for free education or medical treatment.
The farmer, named only as Pierre, said he had not married because he could not afford the bride price demanded by his partner's family.
Hundreds of people have since died since Mr Nkurunziza launched a bid for a third term. But the country had been relatively calm in recent months.
Cohabiting couples in Burundi have until the end of the year to get married or face legal consequences.