Juan Guaidó declared himself interim president last week.
The US and about 20 other countries have recognised him. President Nicolás Maduro retains the support of Russia.
Meanwhile, Attorney General Tarek William Saab has ordered that Mr Guaidó be barred from leaving Venezuela and have his assets frozen.
In a statement on Tuesday, Mr Saab’s office said violent incidents and interference from foreign governments suggested “the commission of serious crimes that attack the constitutional order”.
At least 40 people are believed to have died in recent unrest, the UN says.
Protests have been growing since Mr Maduro began a second term as president this month. He was elected last year during a controversial vote in which many opposition candidates were barred from running, or jailed.
Venezuela has been struggling with a massive economic crisis. Hyperinflation and shortages of essentials such as food and medicine have forced millions to flee the nation.
What has Mr Guaidó been saying?
Speaking to BBC Mundo in Caracas on Monday, he defended his claim to be interim leader.
Mr Guaidó said the Venezuelan constitution allowed him, as the head of the National Assembly, to take power when the president was considered to be illegitimate.
“My duty is to call for free elections because there is an abuse of power and we live in a dictatorship,” Mr Guaidó said.
He added: “In Venezuela, we either accept domination, total oppression and torture… from Maduro’s regime, or we choose freedom, democracy and prosperity for our people.”
Mr Guaidó said the Maduro administration was “killing young poor people” in the streets.
He has pledged to act as interim leader to oversee fresh elections. However, his declaration of leadership has divided the international community.
Mr Guaidó has called for further demonstrations this week, after tens of thousands came out to protest against Mr Maduro last Wednesday.
What is the latest in the diplomatic row?
Russia renewed its criticism of what it calls US interference on Tuesday. It came after US National Security Adviser John Bolton announced sanctions against Venezuela’s state-owned oil firm PDVSA.
Mr Bolton said the move would ensure President Maduro and his allies could “no longer loot the assets of the Venezuelan people”. He again appealed to the military to switch allegiance to Mr Guaidó.
On Tuesday Russian Foreign Minister Sergei Lavrov said the sanctions violated “all possible international norms” in pursuit of regime change.
He also “pledged to do everything to support the legitimate government of President Maduro”.
Russia’s finance ministry said it expected Venezuela to keep up debt repayments it owed.
Meanwhile acting US Defence Secretary Patrick Shanahan declined to say whether the Pentagon was considering sending troops to Colombia, which borders Venezuela.
On Monday Mr Bolton appeared at a briefing with a notepad containing the words “5,000 troops to Colombia”.
“I haven’t discussed that with Secretary Bolton,” Mr Shanahan told reporters on Tuesday.
In New York, Venezuelan ambassador to the UN Jorge Valero accused the US administration of preparing a “military invasion”.
Who supports whom?
Venezuela broke off relations with the US last week in response to its recognition of Mr Guaidó as interim leader and ordered all US diplomats to leave the country.
On Tuesday the state department issued a travel advisory urging US nationals not to go to Venezuela “due to crime, civil unrest, poor health infrastructure, and arbitrary arrest and detention of US citizens”.
More than 20 countries have followed the US in recognising Mr Guaidó as interim president.
Russia, China, Mexico and Turkey are among those who have publicly backed Mr Maduro.
On Saturday, several European countries including Spain, Germany, France and the UK said they would recognise Mr Guaidó as president if elections were not called within eight days.
What are the UN’s casualty figures?
The estimated death toll of 40 includes 26 people shot dead by pro-government forces, five killed in house raids, and 11 who died during looting, UN human rights spokesman Rupert Colville said on Tuesday.
Another 850 people have reportedly been detained since 21 January – including dozens of children as young as 12.