Much of the area is a nature reserve owned by the National Trust, which said it was assisting firefighters.
The area is also close to the RSPB Minsmere nature reserve.
It asked people to avoid Westleton Heath, Mount Pleasant, Westleton Walks and the main bridleway from the entrance road towards Dunwich.
The fire service said crews were using equipment including off-road vehicles, drones and quad bikes.
“Members of the public are advised to avoid the area and keep roads clear for emergency vehicles,” it said.
The fire comes as the UK Health Security Agency has issued an amber heat-health alert for the east midlands, the east, south east and London.
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