Officials say the death toll for the Saturday night landslide at the Koshe landfill is likely to rise. A resident said 150 people were there at the time.
A number of makeshift houses are now buried under tonnes of waste.
The area has been a dumping ground for Addis Ababa's rubbish for more than five decades.
Rescuers are using bulldozers and even bare hands to move tonnes of debris as the search for survivors and dead bodies continues.
At least 10 more bodies have been recovered on Monday.
City authorities say dozens of people are still unaccounted for and could be buried under the rubble.
Dozens of others have been treated and discharged from a local hospital.
Communications Minister Negeri Lencho told the BBC that the government had appealed to the residents to leave the dump:
"We had plans to resettle the people. Unfortunately this landslide occurred in the meantime.
"But now more than 290 people living in the area have been relocated. We plan to support them so that they can live in a safe zone."
Katsela Mengistu, a 50-year-old father of two, is among hundreds of residents who have gathered at the site, waiting and watching anxiously for news about his family:
"I am just here waiting for news of my family members; my wife and two children - a boy and a girl. They are all buried under this landslide. The government is helping so we will just have to wait," Mr Katsela told our reporter.