A state TV reporter who also is a medical doctor issued the dire warning shortly after hard-line Shiite demonstrators pushed their way into the courtyards of two major shrines that were closed because of the virus.
Iran has been hit particularly hard by the virus, just behind China and Italy in terms of the number of cases and deaths. Roughly nine out of 10 of the over 18,000 confirmed cases of the virus in the Middle East come from Iran.
The death toll in Iran saw another 13 percent increase Tuesday. Health Ministry Spokesman Kianoush Jahanpour said the virus had killed 135 more people to raise the total to 988 deaths amid more than 16,000 confirmed cases.
Though most infected people experience only mild or moderate symptoms and recover within weeks, the virus is highly contagious and can be spread by people with no visible symptoms. Older adults and people with existing health problems are at higher risk.
In announcing the new warning, Iranian state TV journalist Dr. Afruz Eslami warned that if people fail to follow any guidance, it could collapse Iran's already-strained medical system. If the "medical facilities are not sufficient, there will be 4 million cases, and 3.5 million people will die," she said.
Underlining that urgency was the fatwa issued by Supreme Leader Ayatollah Khamenei, which prohibited "unnecessary" travel.
Late Monday night, angry crowds stormed the courtyards of the Imam Reza shrine in Mashhad and the Fatima Masumeh shrine in Qom. Many people visit the shrine in Qom 24 hours a day, seven days a week, touching and kissing the shrine. The shrines draw Shiite pilgrims from all over, likely contributing to the virus' regional spread.
State TV had announced the closures earlier in the day, sparking the demonstrations. Police later dispersed the crowds and made arrests. Religious authorities and a prominent Qom seminary called the demonstration an "insult" to the shrine.
The Associated Press contributed to this report.