The Met Police said people were being treated for "a range of injuries".
The BBC's Andy Moore said the whole tower block was alight and there are fears the building might collapse.
At about 04:00 BST police said: "Residents continue to be evacuated from the tower block."
Forty fire engines have been sent to the tower, and Mayor of London Sadiq Khan said a "major incident" had been declared.
London Underground said the Hammersmith and City and Circle lines have been closed between Edgware Road and Hammersmith because of the fire.