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.
Both National Trust Dunwich Heath and the nature reserve have been closed.
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 are 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.
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