A woman in her 80s passed away in the North-West Regional Hospital on Monday morning due to illness related to coronavirus.
Premier Peter Gutwein said the state's first death from the virus was confirmation people needed to understand their isolation, quarantine and social-distancing requirements.
"Unfortunately this won't be the last death," he said.
"There will be more deaths if this gets away from us.
"There will be more families burying their loved ones.
"Now is the time to step up and take personal responsibility, community responsibility, and state responsibility."
Public Health director Mark Veitch said the death of the elderly woman demonstrated the importance of social distancing, isolation and quarantine measures to protect those most vulnerable to the infection.
"Should coronavirus infection become established in Tasmania, which is something we certainly hope to delay and suppress as much as possible, it is an unfortunate fact that the majority of deaths from coronavirus are people in their 70s and 80s," he said.
The national coronavirus death toll stands at 18.
Health Minister Sarah Courtney said two coronavirus cases in the Devonport area to identify possible sources were still under investigation by Public Health Services.
"Neither case had travelled overseas or been on a cruise ship in the past two weeks before their illnesses," she said.
Dr Veitch said these cases were of a particular concern has they might represent the first cases of community transmission of the virus within the state.
He said the testing guidelines for coronavirus would be expanded once there was evidence of community transmission.