Thousands defied a warning from the military council to respect the night-time curfew imposed from 2000 GMT (10 pm) to 0200 GMT (4 am), to maintain their vigil outside army headquarters in Khartoum for a sixth straight night.
Protesters have rejected the military's announcement that it will rule the country for the next two years, calling it a continuation of al-Bashir's regime.
"We have no ambitions at all to hold the reins of power", he told a press conference in the capital, Khartoum.
The new military rulers vowed earlier on Friday to open a dialogue with all political groups on forming a civilian government as protesters railed against their seizure of power after removing president Bashir.
"I announce my resignation from this post and the choice of the one in whose experience I trust... after consultations, Abdel-Fattah al-Burhan was appointed new head of the Transitional Military Council", Ibn Auf said in a televised address broadcasted by Arabic TV channels.
Amnesty has also called for the arrest of al-Bashir for human rights violations during his rule.
Sudan's military ousted al-Bashir on Thursday in response to escalating popular protests.
The Sudanese Professionals Association (SPA), which coordinates the popular protests, also rejected the statement by the defence minister and called on the protesters to continue demonstrating until "comprehensive change" is achieved. He declared a three-month state of emergency, the suspension of the constitution and the dissolution of parliament.More news: DeChambeau, Koepka look to maintain momentum at the Masters
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The former president, who ruled Sudan for 30 years and also came to power through a military coup, has been indicted by the International Criminal Court in The Hague over allegations of genocide in Sudan's Darfur region.
The State Department, while declining to declare the takeover a coup, said it supported a peaceful and democratic Sudan and believed the Sudanese people should be allowed a peaceful transition sooner than only in two years.
Footage shared on social media on Friday claimed to show NISS forces firing on protesters in North Kordofan, in central Sudan.
Lt-Gen Abidin, who heads the military council's political committee, said on Friday: "The solutions will be devised by those in protest".
Ibn Auf said on Thursday that Bashir was being detained in a "safe place" and a military council - which it was later announced he is heading - would now run the country.
He pledged the military would stay only as long as it's needed.
The force draws its origins from the Janjaweed militias that were implicated in the Darfur genocide.
He said that handing over al-Bashir would be "an ugly mark on Sudan. even rebels carrying weapons, we won't extradite them".