For a classic example, consider Keats' "La Belle Dame Sans Merci." It has two tones evident at once because there are two speakers. A knight has "met a lady in the Meads... a faery's child," and he rhapsodizes about their love affair. The knight recounts, "she said, 'I love thee true'/... and I awoke and found me here on the cold hill's side." There is ineffable sorrow and despair in the knight's tone. His companion only notes that it's winter; he is detached from the knight's fate.
In a closer reading, one discovers a phrase that might be overlooked: the lady spoke "in language strange." This realization changes the tone of the knight's speeches; he did not know her language, and her love overtures may have been cries for help. What was desertion to him may have been escape for her. His tone now seems frenzied rather than wistful, and his character self-deceived -- the girl's love may be a fantasy.
Reading yet more carefully yields another nugget of truth: the knight covered his lady with garlands of flowers, but, as the first speaker points out, "the sedge has withered from the lake." The flowers may have been fantasies, which completely upends the knight's tone into one of madness. After all, if the flowers were unreal, the lady may be unreal, and the reader can legitimately doubt the knight's sanity. The first speaker's neutral tone remains unchanged; the knight's tone has been completely altered.
Anyone describing tone in a double-meaning poem should realize that the tone alters, not because the poem changes, but because the emphasis on certain words changes. If one notes tone as an adjective describing a first impression -- this speaker seems worried, nostalgic, happy, sorrowful -- the game becomes to see how the tone changes with a closer study of the poetry. And it often will change, because all great poetry contains layers of meaning. All great poetry also has shifts in tone, achieved through the reader's changing perceptions.