Protesters said they were particularly distrustful of Mr Abnouf, who announced the coup on Thursday and until Friday headed the new military transitional council.
Protest leaders dismissed the transitional military council as the "same old faces" from the old regime, which had led the country into multiple conflicts and worsening poverty and social inequality.
Al-Burhan, who previously served as the general inspector of the armed forces, had met the protesters in the streets after Bashir's removal and listened to their demands.
While recent events in Algeria and Sudan have been unique to those countries, analysts say the rapid downfall Bouteflika and Bashir are a warning to authoritarian leaders in the region that they will ignore popular anger, especially over economic grievances, at their peril.
Sudan's acting United Nations envoy says the military council that took power after overthrowing President Omar al-Bashir "will be the guarantor of a civilian government".
This idea has been rejected by opposition groups who demand a civilian transitional government.
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Auf stepped down on Friday night, in an announcement on national television, and named Lt-Gen Abdel Fattah Abdelrahman Burhan as leader.
"Through consensus among political forces, we will not impose anything on the people", he said during a press conference.
Omar el-Digeir, leader of the opposition Sudanese Congress Party, said the military should not be "the sole custodians of power".
Sudan's constitution was being suspended, border crossings were being shut until further notice and airspace was being closed for 24 hours, he added.
The ousted president had come to power in a coup in 1989 and ruled Sudan for three decades with an iron fist.
Few see him handing over power to a civilian government after the promised two years of transition although he said his government would have 'representation of the people'.
But he said the council would never extradite him, or any other Sudanese, despite a long-standing arrest warrant issued by the International Criminal Court (ICC) against Bashir on charges of genocide and war crimes.
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The demonstrations against Bashir's 30-year rule first erupted in December, triggered by a tripling of the bread prices in one of the world's most impoverished countries. He also vowed to "uproot the regime" of Bashir and its symbols.
The group encouraged continued protests at the army's headquarters in Khartoum and military posts around the country "until state power is reinstated to a civilian transitional government that represents the forces of the revolution".
Earlier on Saturday, the state-run media announced the resignation of Lieutenant General Salah Abdallah Gosh, who as the head of the National Intelligence and Security Services had overseen the crackdown against the protesters.
Protesters want a civilian council to lead the transition rather than a military one.
Sudan Police on Saturday said at least 16 people were killed, and 20 injured by stray bullets at protests and sit-ins on Thursday and Friday. He also said Chief of Staff Kamal Abdelmarouf al-Mahi was relieved of his position as deputy head of the transitional military council.
The Sudanese Professionals Association (SPA) said the military council was "not capable of creating change".
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