The scathing slogans appeared just days after Iran's supreme leader Ayatollah Ali Khamenei said such insults should not be taken personally by US or Israeli citizens.
Tehran's downtown Enghelab, or Revolution Street, is decorated with huge balloons.
Chants of "Death to Israel" and "Death to Britain" followed, and demonstrators burned US and Israeli flags.
In capital Tehran, demonstrators marched across different routes to reach the Azadi (Liberty) Square, where President Hassan Rouhani gave an address.
Tensions between Washington and Tehran are high after the USA withdrew from an global nuclear deal aimed at curbing Iran's nuclear ambitions, and re-imposed sanctions that have hurt the Iranian economy.
Iranians thronged streets across the country, many with portraits of Ayatollah Ruhollah Khomeini, the Shiite cleric who toppled the Shah in the historic Islamic uprising.
"But if they attack us, we will raze Tel Aviv and Haifa to the ground", Yadollah Javani, the Guards' deputy head for political affairs said during a rally celebrating the 40th anniversary of the Islamic revolution.
In the backdrop to Monday's marches, the military displayed Iranian-made missiles, which authorities showcase every year during anniversary celebrations. "Iran continues to seek nuclear weapons to intimidate people all around the globe and ballistic missiles to use as delivery systems". "The regime in Iran has produced only #40YearsofFailure", he wrote.
Mahmoud Hemmati, 35, was pushing his 68-year-old mother, Parivash Fakheri, in a wheelchair.
Fakheri, who was one of the revolutionaries on the streets of Tehran in 1979, said she would defend the revolution all over again. "I don't think you'll have many more anniversaries to enjoy", Bolton concluded.
"40 Years of Failure to accept that Iranians will never return to submission".
A pre-prepared resolution was read out ahead of his speech that proclaimed "unquestioning obedience to the supreme leader Ayatollah Ali Khamenei" and called U.S. President Donald Trump an "idiot".
Trump was echoed by National Security Adviser John Bolton, who published a spate of tweets, in which he argued in particular that "the 40th anniversary of the Iranian regime only serves to highlight four decades of failure and broken promises".
Last year, the government suppressed protests over poor living conditions in over 80 cities and towns.