Up to 200 people are missing, the fire brigade says, although there have been no confirmed fatalities.
Rescue teams were dispatched and officials said a number of residents close to the dam had been evacuated.
The dam and Feijão iron ore mine are owned by Brazil's largest mining company, Vale.
Vale and local authorities said on Friday that leaked tailings from the Corrego de Feijão mine had spread into the nearby community Vila Forteco, close to Brumadinho.
Footage broadcast on local television showed the mudflow sweeping over roads and destroying buildings in its path.
Inhotim, Brazil's main contemporary arts centre, is in Brumadinho, but is not believed to have been affected.
A government spokesperson said Brazil's environment minister, Ricardo Salles, was travelling to the scene.
President Jair Bolsonaro will visit the region on Saturday.
The incident comes four years after another dam - owned by Vale and BHP Billiton - burst in Minas Gerais, killing 19 people.
It is considered Brazil's worst environmental disaster.
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