If we read Sappho's "Hymn to Aphrodite" as having a moral or message at all (as opposed to being a simple prayer or a description of the pains of loving), then it is to be found in Aphrodite's reassuring response to the speaker's petition. The goddess of love says that the one who runs away will soon follow her; the one who rejects her gifts will soon be giving them and will love her even if she no longer wants that love.
The moral here is that love is fickle and transitory. Today, the speaker burns with desire for one who disdains her. Tomorrow, these two positions may be completely reversed.
The fact that Aphrodite brings the speaker this comfort, of course, can be said to bear a moral about the importance of piety and paying respect to the gods if one is to be healed. Indeed, the verb θεραπεύω, from which the English word "therapy" is derived, means both to serve the gods and to heal sickness.