What are some examples of works we read today as poems, but were originally sung, their melodies lost over time?Is a song a poem? What poems were...

You are referring to oral epics. These are commonly-known stories, transmitted from folk to folk through song in absentia of a printing press, literacy, or in need to confer information quick. In this case, oral epics were, of course, told from folk to folk in song in order to a) say the same story twice  b) remember facts  c) nobody could have written it by then.

In Spanish literature you have El Cid Campeador, which is an epic tale just like those in the likes of Beowulf *also once a folk-to-folk sung epic*. The Iliad and the Odyssey were also oral epics sang in song or told like stories.

Some recent examples, however, can be found in African American history and the slave lore, which was nearly entirely based on chant and song.

I am including some links for you to see a diversity of lists available online. Hope it helps!