Rallies replete with "Death to America" chants are staged on the embassy takeover anniversary every year, but US-Iranian rancour is especially strong this year following Trump's decision in May to withdraw the United States from a 2015 worldwide nuclear deal and reimpose sanctions on Tehran.
Hardline students stormed the embassy on November 4, 1979, soon after the fall of the USA -backed shah, and 52 Americans were held hostage there for 444 days.
The seizure of the USA embassy by radical students was a key stage in the Islamic revolution of 1979, leading to a 444-day hostage crisis that permanently damaged relations between Washington and Tehran.
State media said millions turned out for rallies in towns and cities, swearing allegiance to the clerical establishment and Supreme Leader Ayatollah Ali Khamenei. The turnout figure could not be independently confirmed by Reuters.
"Mr. Trump! Do not ever threaten Iran", Major General Jafari said in a gathering of demonstrators marking the anniversary of United States embassy takeover in Tehran on Sunday.
After unilaterally pulling out of the deal meant to curb Iran's nuclear capabilities earlier this year, the White House said it would reapply sanctions on the country's energy, financial and shipping sectors.
Trump said the deal was weak and favoured Iran. The other signatories remain committed to the accord.
Iran has stuck with the deal, but the USA pulled out unilaterally saying the agreement did not go far in enough to curtail Tehran's regional involvement, including support for Bashar Assad in Syria.
On Fox, Pompeo called the penalties "the toughest sanctions ever put in place on the Islamic Republic of Iran" and defended the U.S.' decision to give eight unidentified countries more time to wean themselves off Iranian crude oil.
The Trump administration has worked to crush Tehran's oil exports to zero since May, when Mr. Trump withdrew the US from the multilateral deal endorsed by the Obama administration that eased global sanctions in exchange for curbs on Iran's suspect nuclear programs.
The restoration of USA sanctions on Monday targeting Iran's oil sales and banking sector is part of an effort by Trump to force Iran to halt its nuclear and ballistic missile programmes outright, as well as its support for proxy forces in conflicts across the Middle East. "But America's plots and its plans for sanctions will be defeated through continued resistance", said Jafari. "The whole world understands these sanctions are real", he said during an interview of CBS's Face the Nation.
He said "economic warfare" was a final bid by Washington to overthrow the Islamic republic after decades of failed attempts.