Since the beginning of the week, clashes between M23 rebels and the army and its allies left hospitals overwhelmed by casualties and bodies on the streets, according to the UN.
Warehouses with food and medical supplies were also looted, aid agencies said.
Essential services have been unavailable since the offensive began, with electricity, water and internet cut.
Growing anger over the rebel offensive led to protesters targeting foreign embassies in the capital, Kinshasa.
At least 10 embassies were damaged and looted by protesters who demanded that the international community intervene to end the rebel assault, as they tightened their grip on Goma.
The RDF said the Romanian mercenaries were being transported to Rwanda’s capital, Kigali, after they “surrendered to M23 following the capture of the strategic city”.
AFP news agency reports that dozens of Romanians – whom it called private military contractors – had fled to the Rwandan town of Gisenyi, just over the border from Goma.
“We weren’t on a battlefield, we were here to train and help with artillery,” one who only gave the name Emile told AFP.
“What I learned is that anything can happen. I came. I did my work and now I am happy to go back and be with my family,” he added.
In an apparent attempt to extend their territorial control in eastern DR Congo, M23 rebels were now advancing towards the lakeside city of Bukavu, Reuters news agency quoted five diplomatic and security sources as saying.