How does the tone change throughout the poem "O Captain! My Captain!"?

The tone of "O Captain! My Captain!" does not so much change over the course of the poem as it maintains a tension between exultation and melancholy throughout its three stanzas. In the first stanza, we see this in the shift in language between the first half—"prize," "won," "people all exulting"—and the second, which marks a semantic and emotional step change away from the "daring" vessel towards the "bleeding" heart of the poet, as it is revealed that the Captain is "cold and dead."
Whitman repeats this effect throughout the next two stanzas. At the beginning of each, he describes the scenes of victory with which the captain is greeted: "the flag is flung," "the bugle trills," "the shores a-crowding," but once more the refrain appears that the Captain is "fallen cold and dead."
This repetition—not strictly a refrain, but epistrophe—emphasizes the continuing use of the same technique, the contrast of triumph with sadness which the poet deeply wants the reader to feel. In the final stanza, the focus is on the Captain for the entirety of the stanza, but still, there is a discrepancy between the "voyage closed" and "object won" in the first half, and the piteous "fallen cold and dead" in the second. As befits the subject of the poem, it is this refrain that the reader is left with, as well as this sense of tension between triumph and despair.