The Portuguese coach, whose side lost all three matches at the tournament co-hosted by Equatorial Guinea and Gabon, follows Amara Traore of Senegal on the list of coaches to pay a price for failure at the Nations Cup.
Burkina Faso Football federation president Zembende Theodore Sawadogo told a weekend radio interview he could see no reason to continue with the 42-year-old, who had been Burkina Faso coach for almost four years and also took the side to the 2010 Nations Cup finals in Angola.
Duarte proved a controversial figure during his tenure, accused of persuading African players he knew from his time coaching in Portugal to accept offers of Burkina Faso nationality to strengthen his side.
A case against the Cameroon defender Herve Zengue, who played for Burkina Faso in the qualifiers, almost cost the country their place at the Nations Cup.
Duarte also incurred widespread derision last June when he tried to persuade 35-yaer-old Brazil-born goalkeeper Nilson of Vitoria Guimaraes in Portugal to play for Burkina Faso. It was an idea that was eventually dropped.
Burkina Faso will have to have a replacement in place before June when they begin World Cup qualifying with matches against Congo and Gabon.