A health worker and male patient at the North West Regional Hospital in Burnie returned positive tests to COVID-19 on Sunday.

Three workers at the facility have previously tested positive.

Health authorities are trying to trace the source of the cases and are contacting recently discharged patients to determine if they have symptoms.

"Whilst it is concerning, this will not be the last time that we will see this type of outbreak at a hospital in Tasmania," Premier Peter Gutwein said on Monday.

"Importantly for those people that live on the northwest coast, the message could not be stronger ... take into account the rules relating to social distancing."

The patient had been transferred on April 2 to the Mersey Community Hospital, in Latrobe in the north, before showing symptoms.

Six staff at the NWRH and 21 at the Mersey Community Hospital are in quarantine.

Transfers between the hospitals have been stopped while health authorities are urging discharged patients to contact the public health hotline if they feel unwell.

The state's first coronavirus death, an elderly woman, occurred at the NWRH last month.

Mr Gutwein implored people not to travel unnecessarily in the state and did not rule out locking down cities or regions if needed.

Three new cases, including two in the northwest, were confirmed on Monday night, taking the state's overall tally to 89.

Tasmania has had two COVID-19 deaths while 29 people have recovered.

The state government recently tightened restrictions around visiting holiday homes and taking boating trips over the Easter period.

Mr Gutwein also flagged stricter retail and supermarket restrictions if social distancing rules aren't followed.

"If we don't get this right ... then I will have no other option than to shut you down," he said.

"I don't want to do that. But if that's where we need to get to and we need to identify only essential retail, then that is what we will do."

Australian Associated Press