One resident, Amal Hussein, said she saw "police cars surround" the school and heard people screaming.
"Police came and ordered us to leave the school, based on a decision from the governor, and fired tear gas at us," Hussein Gomaa, who had been displaced from the capital, told AFP.
"We are 770 people who had fled the war in Khartoum and were sheltering in this school," Gomaa said after fleeing the makeshift displacement camp, where he said hundreds of people "had been receiving aid."
"We don't understand why we were driven out," he said. "Now we're out in the open with women and children, and we don't know where to go."
Gedaref currently hosts 273,000 people uprooted in the conflict between the Sudanese Armed Forces, SAF, and the paramilitary Rapid Support Forces, RSF.
According to the United Nations, thousands are being housed in makeshift camps such as schools where food, clean water and health care are in short supply.
Barely two hours after they were forced out, Suleiman Mohammed, who had also been sheltering at the school, said they were again "evacuated from the dormitories" of Gedaref University's medical school.
"Police said the decision was issued by the governor," he added.
In Khartoum, a committee of volunteers reported "intensified clashes" in a densely populated northern neighbourhood.
Since April, forces loyal to army chief Abdel Fattah al-Burhan — Sudan's de facto head of state — have been at war with the RSF commanded by his former deputy, Mohamed Hamdan Daglo.
More than 10,000 people have been killed, according to a conservative estimate from the Armed Conflict Location & Event Data project.
The warring sides failed to agree on a cease-fire in a new round of Saudi- and U.S.-brokered talks this week in Jeddah.
In the vast western region of Darfur, where some of the worst fighting has taken place, the RSF has claimed control of all but one major city.
Their advance amid a communications blackout has triggered renewed fears of ethnically motivated mass killings.
In Washington, State Department spokesman Vedant Patel voiced concern about "extreme fighting" in parts of Sudan.
"The U.S. condemns in the strongest terms reports of killings on the basis of ethnicity by the RSF and its allied militias in West Darfur," he told reporters.
Of 4.6 million people internally displaced within Sudan, more than three million people have fled the violence in Khartoum, according to UN figures.
Sudan is facing an "unimaginable humanitarian crisis," the UN refugee agency said Tuesday, with most hospitals shuttered and millions in severe need of aid as the violence continues unabated.