A spokesman for the UN human rights office urged security personnel not to use excessive force, including firing live ammunition, against demonstrators.
He also called on Iran’s government to restore access to the internet.
The government has blamed the turmoil on foreign enemies and denounced those on the streets as “rioters”.
The current situation is unclear because of the near-total internet shutdown, which started late on Saturday, but protests have been held in dozens of cities and towns.
The protests erupted on Friday after the government announced the price of petrol would be increased by 50% to 15,000 rials ($0.12; £0.09 at the unofficial market exchange rate) a litre and that drivers would be allowed to purchase only 60 litres each month before the price rose to 30,000 rials.
President Hassan Rouhani said the government was acting in the public interest, and that the money raised would be distributed to the country’s neediest citizens.
However, the decision was met with widespread anger in a country where the economy is already reeling as a result of US sanctions reinstated last year when President Donald Trump abandoned a nuclear deal with Iran.
The sanctions have caused Iran’s oil exports to collapse and the value of the rial to plummet, and sent the prices of basic goods soaring.
By Sunday, the protests had reached some 100 cities and towns, the semi-official Fars news agency said. At least 100 banks and 57 shops had been set on fire, and about 1,000 people had been arrested, it added.
A government spokesman said the situation was “calmer” on Monday and predicted that “tomorrow and the day after we won’t have any issues with regard to riots”.
However, videos posted on social media showed protests continuing in several locations on Monday night. Three security personnel were also stabbed to death near the capital Tehran, local media said.
On Tuesday, a spokesman for the office of the UN high commissioner for human rights told a news conference in Geneva that it was deeply concerned by reported violations of international norms and standards on the use of force, including the firing of live ammunition, against demonstrators.
Rupert Colville said it was extremely difficult to verify the overall number of deaths, but that Iranian media and a number of other sources suggested that dozens of people might have been killed during protests in at least eight different provinces.
“We urge the Iranian authorities and security forces to avoid the use of force to disperse peaceful assemblies and in cases in which an assembly is violent to restrict the use of force to the greatest extent possible.”
“We also urge protesters to carry out demonstrations peacefully, without resorting to physical violence or destruction of property,
Mr Colville also called on the Iranian government to respect the rights to freedom of expression and peaceful assembly, and to immediately re-establish Iranians’ access to the internet.
He said the UN noted the serious economic challenges Iran was experiencing, in part because of the US sanctions. But he warned that simply responding to the protests “with harsh words and an iron first” only risked making the situation worse.
“Protests of this nature and on this scale are an indication of deep-rooted and often well-founded grievances, that cannot simply be brushed aside,” he added.
President Rouhani has insisted that insecurity cannot be allowed.
“People have the right to protest, but protests are different from riots,” he said on Sunday. “We should not allow insecurity in the society.”