If the Germans really follow the Hitler plan and hold the ring and shoot anybody who tries to surrender, I suspect the civilian polulation would go north to Finnish lines and surrender there. This will lead to some immediate squabbling since the Finns will need help to feed these people and might just send them all to some point on their front and return them to the Soviets. In the winter without food it will be the death march for sure, and everybody in the world will soon know.
Finland has no way of feeding or housing a major number of Soviet civilians, even if the government wanted to. IOTL, Finns did not have the resources to adequately feed or house even the Soviet POWs that were captured during the attack and the death rates in POW camps '41-'42 were horrible. It was partly racism and poor organisation, too, but also for some parts of the Finnish civilian population the conditions during that winter were close to famine.
So, yes. A huge part of the Soviet civilians who seek refuge in Finland would die in Finland, because of a combination of malnutrition, cold and disease in makeshift camps. Even if the Finns would genuinely try to help them.