In order to understand what Lincoln is saying in this line, you need to look further down in the speech. Lincoln goes on to tell us what the "unfinished work" is. Basically, the work that the living must continue to do is the work of making sure that democracy can exist.
In the speech, Lincoln says that the Civil War is a war for democracy. He says that it is a test of whether a government that is
... conceived in liberty, and dedicated to the proposition that all men are created equal
can last for any long period of time. The men who have died in the battle, then, have given their lives to try to ensure that democracy can last.
According to Lincoln, the living need to work to preserve democracy. They need to continue where the dead left off. If they do they will be able to ensure
that government of the people, by the people, for the people, shall not perish from the earth.
So, the "unfinished work" is the work of keeping democracy alive and well.