Some 3,000 people have been left homeless, according to a BBC reporter who has been to the scene.
Their houses were destroyed after mud and rocks, dislodged by the rain, plunged down the hillside.
Reports from the Democratic Republic of Congo say there have been similar problems across the border.
UN-backed Radio Okapi says 14 people have been killed on the Congolese side of Lake Tanganyika.
Burundi's President Pierre Nkurunziza has visited the scene of the landslides, 35km (20 miles) south of the capital Bujumbura.
The governor of Bujumbura district, Jacques Minani, described it as "a disaster", reports the AFP news agency.
He said that the emergency services were now searching the area for survivors.
At least 10 people are missing, feared dead, in Burundi after landslides which have destroyed hundreds of homes, officials say.