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...