It is believed the cause of the huge blaze was connected to a toaster on the ninth floor of the building, reports the Belfast Telegraph.
Julie Ann Jackson, from the Seymour Hill and Conway Community Residents - which is helping those who live in the flat - gave details about the blaze.
She said the fire started in the flat of man aged in 50s who lived in the building on his own and he was luckily able to inform other residents about the fire.
Ms Jackson said: "I think they were shook up from Grenfell, and after tonight they are really shook up."
The flats have been evacuated and it is believed the fire broke out on the ninth of tenth floor of the building.
Locals could be seen crowding outside the tower on Seymour Hill and people have been posting on social media.
David Holt wrote on Twitter: "That fire in the highrise flats in Dunmurry looks terrible! Hope everyone is ok!"
Charlie McLaughlin added: "Seeing fire services racing through Dunmurry to get into this building. Different breed....bravery beyond my comprehension."
One community worker said the fire is now "under control".
It is unclear what started the fire or whether cladding had been added to the building's exterior.
The Northern Ireland Fire and Rescue Service are currently at the scene.
A spokesman for the Police Service of Northern Ireland said the matter was being dealt with by the fire service.
A spokesman for the Northern Ireland Ambulance Service said: "We received a call at 5.40pm from the Northern Ireland Fire Services to reports of a fire on the ninth floor of Coolmoyne House.
"Four Accident and Emergency crews, five rapid response paramedics, seven ambulance officers, one doctor and two heart paramedics attended the scene.
"Four people were led to safety by the Northern Ireland Fire Service and handed over to the Ambulance Service for assessment.
'"One person has been taken to hospital with minor injuries and the three others are still being assessed at the scene."
