The 29-year-old, who was pulled over on Tuesday night in Launceston, is accused of breaching public health directions by being away from his primary residence without a lawful reason.

He was charged with resisting arrest after allegedly telling officers he had the virus.

The man appeared in Launceston Magistrates Court on Wednesday on charges including breaching a family violence order, evading police and driving unlicensed.

Tasmania is strictly policing orders that prevent people from leaving their home unless for essential tasks, work or school, exercise, medical reasons or when providing compassionate care.

Fines of up to $16,800 or three months' jail are on the cards for those who don't comply.

Premier Peter Gutwein has bluntly warned people to not be complacent after the island state went a day without recording a new COVID-19 case.

Two elderly people have died from the virus this week, but no positive cases were confirmed on Tuesday - the first time in about a fortnight.

Mr Gutwein said the number of cruise ship passengers and overseas travellers arriving in the state was easing amid tight border restrictions.

But he said policing of quarantine and social gathering orders would not ease.

"If this was a game of football, we would not even be through the first quarter yet," Mr Gutwein told reporters.

"This is serious. It is going to need us to play a long game."

Tasmania had 69 cases of COVID-19 as of Wednesday afternoon, 11 of whom have recovered.

More than 2200 tests have been carried out, with about 200 done on Tuesday.

The state government has halted renter evictions for 90 days, after emergency legislation passed though parliament last week.

"This is not a licence not to pay rent. If you can pay your rent, you should. That is the very clear expectation," Mr Gutwein said, flagging similar safeguards for commercial tenants.

Financial relief on energy bills was also announced on Wednesday.

While small business customers and community service organisations will have their next quarterly energy bill waived, all energy consumers will see energy prices capped for a year.

The government also said Aurora Energy will not disconnect any residential or small business customers without their agreement.

Health authorities continue to investigate two virus cases in the Devonport area, in the northwest, where the source of infection is unknown.

School attendances have dropped to below 10 per cent as parents opt to keep children home.

Australian Associated Press