"As humans gifted with intelligence," or "As humans are gifted with intelligence,"?

Solution 1:

Your teacher is wrong, the word are is not grammatically necessary here, and it does change the meaning.

In your sentence, the group gifted with intelligence is a subset of all humanity, you claim membership in the subset, and you assign an obligation to the subset.

In your teacher's sentence, the group gifted with intelligence is all humanity, and you assign an obligation to the entire species.

Your teacher's version is probably better (expressing that there may be non-intelligent humans may offend), but your version is not ungrammatical, and because your version remains correct even when the subset is not strict (i.e. it contains the entire species), you aren't actually making the offensive claim, just leaving the possibility open.