Tennyson's poem "Crossing The Bar" has been set to music:
Lyr Add: Crossing the Bar (Tennyson, Arbo)
I believe that thread has a link to Ms. Arbo singing that song on Prairie Home Companion.

And the here's a thread with the tune: Tune Req: Crossing the bar - Tennyson

Emma Lazarus's poem "The New Colossus," which is the inscription on the Statue Of Liberty, has also been set to music, but I don't know who has recorded it.

One of the best known poems set to music is Longfellow's "I Heard The Bells On Christmas Day," which is sung to at least 2 different tunes.   Both song versions are easy to find in books of Christmas carols or hymns.
The song versions, which I believe first came out about 10 years after Longfellow wrote and published the poem, leave out two of the verses with specific allusions to the cannons of the War Between The States.   (It was the juxtaposition of the cannons' roar with the Christmas day bells that inspired Longfellow to write the poem.) And the verse order is rearranged in the song, too.