There are reports of the army using heavy artillery in the central province of Gitega, and fierce battles in the east and the north-west.
The BBC's Christophe Nkurunziza in the capital, Bujumbura, says that the fighting has escalated as a meeting between President Pierre Buyoya and FDD leader Pierre Nkurunziza approaches.
He says both sides are trying to capture as much territory as possible ahead of peace talks which are scheduled to resume in Pretoria within the next few days.
The violence has continued despite both sides signing a ceasefire last December.
Looted
Some of the people who have fled in Gitega are sheltering in Catholic churches outside the towns where they live, and thousands of others have travelled many miles from the fighting.
Others have fled to neighbouring provinces or even across the border to Tanzania.
Among the many attacks in Gitega province, some 40 gunmen, alleged to be FDD rebels, raided a district in the main city of Bugendana, looting people's homes.
A local resident was severely wounded after trying to cry for help.
Tension is also mounting in the neighbouring province of Cankuzo, where the authorities say the rebels have been getting reinforcements from across the border, in Tanzania.
Army reform
Neither the army nor the rebels have provided casualty figures, but there are reports of civilians killed in the east of the country.
Earlier this week, 10 soldiers are believed to have died in a rebel ambush.
More than 300,000 people, mainly civilians, have been killed in the 10-year civil war in Burundi.
The ethnic Tutsi minority has traditionally dominated Burundi.
The talks in South Africa are set to tackle issues such as restructuring the army and giving cabinet posts to FDD leaders.