In the poem "Tears, Idle Tears" the designation "idle" is a bit obscure, because while the poet opens with the expression "Tears, idle tears, I know not what they mean," the main theme of the poem does not seem to be tears specifically, but rather regret and longing for the past.
The word "idle" typically refers to something that is inactive, indigent, stagnant, useless, or deliberately not fulfilling its designated occupation. In the poem "Tears, Idle Tears," the word "idle" seems to most closely fit the idea of "useless"—that is, something that exists but serves no purpose.
To understand the phrase "tears, idle tears" in this poem it is necessary to carefully consider the poem in its entirety.
Consider the ideas that are repeated in this poem. "Tears... rise in the heart... thinking of the days that are no more" (paragraph 1); "So sad, so fresh the days that are no more" (paragraph 2); "So sad, so strange, the days that are no more" (paragraph 3); "O Death in Life the days that are no more!" (paragraph 4).
Through these phrases it becomes apparent that the main theme in this poem is the poet's experience of deep sadness when considering the past—a grief that makes life itself feel like death (paragraph 4). The poet's tears, however, are "idle"—that is, they are useless, they are of no benefit. He grieves over the past, which by nature and definition cannot be changed or affected by tears or any other actions.