
The suspect came from Myanmar and was known within the Melbourne Rohingya community, Habib, a member of the Australian Burmese Rohingya Organization said.
Rohingyas are a Muslim minority group in Myanmar who are subjected to persecution and violence from the country's Buddhist majority.
"It is very shocking for us, but this is not about race or religion. This is about the mental health condition of an asylum seeker," Habib said.
Police were yet to reveal the identity of the suspect or his motive.
The man arrived in Australia by boat in 2013 and was living alone in a Melbourne suburb while he was waiting to be accepted as a refugee, Habib said.
He set himself on fire Friday morning inside the Commonwealth Bank branch in Springvale, in Melbourne's south-west, before being arrested and placed under police guard in hospital.
Many asylum seekers were in a legal limbo as the Australian immigration department was too slow to process their applications, causing mental anguish and depression, Habib said.
In recent months the man faced financial difficulties as he had stopped receiving government benefits.
There are approximately 600 Rohingya asylum seekers currently living in the Melbourne area.