It doesn't make sense unless you expand the quote a little:

But, in a larger sense, we can not dedicate -- we can not consecrate -- we can not hallow -- this ground. The brave men, living and dead, who struggled here, have consecrated it, far above our poor power to add or detract. The world will little note, nor long remember what we say here, but it can never forget what they did here. It is for us the living, rather, to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced.

Lincoln explained that there was no point for those who were alive (i.e. hadn't died in the battle) to even try dedicating, consecrating, etc., since the world wouldn't wouldn't care much what was said. Instead ("rather"), they ("the living") needed to focus their efforts on making sure that the soldiers killed in that battle hadn't died in vain; that instead of saying things, they (the living) should be doing things, i.e. preserving freedom and government of the people, by the people, of the people.

"For us" in this context means, roughly, "up to us."

As in "It's for you to decide where you want to study English."

The construction is for + noun/pronoun + infinitive.

1.- We use this structure when an infinitive needs its own subject.
— It's for Jane not to fail her exams. (Jane is subject of the verb fail.)

2.- It is frequently used after adjectives and nouns.
— I'm unhappy for the children to miss school.

3.- It can be the subject of a sentence.
— For him to apologise would be unthinkable.

4.- It is often used in place of a that-clause with should or the subjunctive and is less formal.
— It's for Sue to ring her dad.