London Fire Brigade said at least 120 firefighters tackled the blaze, now under control, from 15:45 BST.
Hammersmith and Fulham Council tweeted: “Rest areas and housing are being arranged for those affected”.
London Ambulance Service said it treated two patients at the scene, but neither required hospital treatment. One person was treated for smoke inhalation, LFB said.
‘Staying with friends’
Fire crews are now investigating the cause of the blaze.
Ella Zwart, a talent acquisition consultant, who lives on the 15th floor, has told BBC London she was at work when the fire took hold.
She does not know yet if she can return to her home tonight, but said she was “hugely impressed” with the fire crews who battled the blaze.
She said: “I’m grateful for my support network here in London and happily staying with friends for the time being.”
The fire was under control by 17:32 BST but crews remain at the scene and have been “damping down”.
The blaze damaged around a quarter of a four-room flat on the seventh floor, half of a flat on the eighth floor, around a quarter of a flat on the ninth floor, half of a flat on the tenth floor and a small part of the flat on the eleventh floor, LFB said.
Shadwell fire station manager Paul Hobbs said: “”The fire spread from the seventh floor via the outside of the building.
“Crews wearing breathing apparatus used jets to extinguish the fire on each floor of the building. They worked quickly in difficult conditions to tackle the fire.”
At the height of the blaze, nearby Shepherd’s Bush station was closed and the adjacent West 12 shopping centre evacuated.
Shepherd’s Bush Green remains closed in both directions, leading to queues in surrounding areas including Holland Road and Uxbridge Road.
Wood Lane is also closed southbound from its junction with South Africa Road.
Witness Helena Noifield told BBC Radio London: “Everyone’s standing around not knowing where to go.
“It’s all pretty horrible really. There’s ash everywhere, smoke everywhere… it’s like something out of a horror film. It’s ghastly.”
An air ambulance and several ambulance crews were on Shepherd’s Bush Green.
Natasha Wills, assistant director of operations at London Ambulance Service, said: “We sent multiple resources including ambulance crews, single responders in cars, our hazardous area response team and an incident response officer.
“We treated two patients on scene, but they were not taken to hospital. We remain at the scene.”
Fire crews from Hammersmith, Kensington, Acton, Chiswick, Paddington and Fulham were at the scene. The cause of the fire is not known at this stage.