Are there some instances in which a poem's title might not be that important? How do you decide whether you need to look closely at the title?
That is a good question. A good starting point is to ask whether the title was assigned by the author or by a later editor.
Titles were generally not important in antiquity and the middle ages. Authors rarely assigned titles to their own poems; most of the titles we have were created by later editors (e.g. Sappho's "Ode to Anactoria"). Works were generally referenced by their incipit (first line) or position in a collection (e.g. Catullus XVI is the 16th poem of Catullus in the standard modern edition). Even as late as the Elizabethan and Jacobean period, individual short poems did not always bear titles (we reference Shakespeare's sonnets by number -- e.g. Sonnet 29)
Poetic titles became more important with the rise of periodical publication in the 18th century, as this was when poems started to by circulated beyond the realm of acquaintances of the poet in the form of individual poems in magazines (earlier, poems were published in books or circulated individually to friends or patrons).
Periodical publication meant that the title of a poem would appear in a table of contents and was used to draw the reader's attention. Thus in western poetry, in general, titles become important in 19th century and later.
Some poets take more care with titles than others, though, and to a degree, you still need to judge based on the practice of the individual poet.