Officials had yet to confirm if anyone had been killed, but Tehran mayor Mohammad Bagher Ghalibaf told state television that between 20 and 25 firefighters were thought to be inside when the building fell.
Three people were pulled alive from the rubble and taken to hospital, the head of Tehran's emergency services said, according to the ISNA news agency.
At least 70 firefighters were injured and 23 hospitalised as they rushed to evacuate the building, which had been on fire for around four hours before it crumbled, the head of emergency services told state television.
Dramatic images showed flames pouring out of the top floors just before it came down live on television.
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"I was inside and suddenly I felt the building is shaking and is about to collapse. As we gathered colleagues and got out, a minute later the building collapsed," said Ali, a firefighter at the scene.
Local media said the workshops were especially full of clothes in the build-up to Nowruz, the Iranian new year, which falls in March.
"More than 30 times we warned the building's owners that it was not safe, but unfortunately they did not pay attention," said Shahram Gilabadi, spokesman for the municipality.
Fire brigade spokesman Jalal Malekias said the building was known to breach safety standards.
"Even in the stairwells, a lot of clothing is stored and this is against safety standards. The managers didn't pay attention to the warnings," he told state television, adding that it lacked sufficient fire extinguishers.
The steel skeleton of the building could be seen bending down to the ground as around 100 fire engines and dozens of ambulances surrounded the area.
Firefighters caked in soot wept outside the building as the hunt for their colleagues continued, and local media showed Tehranis queueing to donate blood.
"I've lost my entire stock. Thousands of families have been ruined," said Ahmad, the owner of one of the 400 shops and business units.
The Plasco building was the first high-rise and shopping centre in Tehran and was the city's tallest building when it was finished in 1962, before being dwarfed by the construction boom of later years.