The army is now being deployed to help set up heat stroke centres, with temperatures reaching 45C (113F).
Officials have been criticised for not doing enough to tackle the crisis.
There is anger among local residents at the authorities because power cuts have restricted the use of air-conditioning units and fans, correspondents say.
Matters have been made worse by the widespread abstention from water during daylight hours during the fasting month of Ramadan.
On Tuesday, the National Disaster Management Authority (NDMA) said it had received orders from Mr Sharif to take immediate action to tackle the crisis.
This came as Sindh province Health Secretary Saeed Mangnejo said 612 people had died in the main government-run hospitals in the city of Karachi during the past four days. Another 80 are reported to have died in private hospitals.
Many of the victims are elderly people from low-income families.
Thousands more people are being treated, and some of them are in serious condition.
Hot weather is not unusual during summer months in Pakistan, but prolonged power cuts seem to have made matters worse, the BBC’s Shahzeb Jillani reports.
Sporadic angry protests have taken place in parts of Karachi, with some people blaming the government and Karachi’s main power utility, K-Electric, for failing to avoid deaths, our correspondent adds.
The prime minister had announced that there would be no electricity cuts but outages have increased since the start of Ramadan, he reports.
Karachi resident Iqbal told the BBC on Monday that no-one in his family could go outside to work because of the temperature and that everyone in their area preferred to stay at home.
“In our area, there is no electricity [since the] morning. We have complained several times, but there is no response from K-Electric,” he said.
According to Pakistan’s meteorological office cooler weather is forecast from Tuesday.
The all-time highest temperature reached in Karachi is 47C, recorded in 1979.
Last month, nearly 1,700 people died in a heatwave in neighbouring India.