On Lesbos itself there are in excess of 5,600 people, more than 2,000 more than the capacity of the camps.
Brawls are common, with many desperate to avoid being returned to Turkey or their home countries after spending a small fortune and risking their lives to escape poverty and persecution.
The fire comes as UN member states Monday promised to try to improve the plight of millions of refugees around the world.
Speaking at the first UN refugee summit in New York, Greek Prime Minister Alexis Tsipras warned that failure to confront the refugee crisis would unleash xenophobia.
“If we fail to support this, the political repercussions will be felt not only in Greece but everywhere,” he said.
“We will give space to nationalistic, xenophobic forces to show their face for the first time since the Second World War.”
More than 850,000 migrants arrived on the Greek islands last year, many after risking their lives in unseaworthy boats and dinghies.