There’s an old Irish drinking song called “Maloney Wants a Drink.”
The lyrics are here:
http://www.kinglaoghaire.com/site/lyrics/song_483.html
All through history, and all through the Bible, Patrick John Maloney could be found seducing women and getting drunk. The songs alludes to Maloney being in the Garden of Eden with Adam and Eve, and to Salome doing her dance of the 7 veils for him.
Bob Marley’s “Survival” alludes to Daniel in the lion’s den, and the miraculous survival of Daniel’s friends Shadrach, Meschach and Abedneg in the fire.
The narrator in David Olney’s Jerusalem Tomorrow is ready to observe–or take part in–a pretty big New Testament event.
(Emmylou covered the tune.)
I had the same problem with John Prine’s Jesus: The Missing Years. 
Forgot one: Creeping Death by Metallica is about the Hebrew plagues against Egypt.
All women are bad, by the Cramps. The first half of the song is all bible stories. http://www.metrolyrics.com/all-women-are-bad-lyrics-cramps.html
The Residents’ entire Wormwood album.
Samson & Delilah by the Grateful Dead
Actually the Dead have plenty of songs that make Biblical references…
would rule out
and
…both being covers of old folk spirituals.
“Man Smart (Woman Smarter)” recorded by Harry Belafonte et a whole lotta al.
Samson was the strongest man long ago
No one could a beat him, as we all know
Until he clash with Deliah on top of the bed
She told them all the strength was in the hair of his head
Also a verse about the Garden of Eden, and another about Methuselah, which I won’t reproduce here to avoid quoting nearly half the song.
How about “Rock Steady” by Sting?
“God Gave Rock n Roll To You” by Kiss. Because on the eighth day, God created…
Not to be confused with Samson & Delilah by Bad Manners
More Cohen: Last Year’s Man
Some women wait for Jesus and some women wait for Cain.
So I hang upon my altar, and I hoist my ax again.
And I take the one who finds me back to where it all began,
When Jesus was the honeymoon and Cain was just a man.
There are some in Supper’s Ready by Genesis
Last verse:
There’s an angel standing in the sun, and he’s crying with a loud voice,
“This is the supper of the mighty One”,
The Lord of Lords,
King of Kings,
Has returned to lead His children home,
To take them to the new Jerusalem.
Toy Matinee’s Last Plane Out starts with the line, “Greetings from Sodom” and continues in that theme. More directly, the second verse begins:
There was one repressed do-gooder
And a few who still believed
Yes I think there were five good men here yesterday
But they were asked to leave
Don’t forget the line “she cut your hair” in Cohen’s Hallelujah.
The Burning Sensations had the song Belly of the Whale (with the chorus “I feel like Jonah in the …”).
edit:
Just recalled U2’s Until the End of the World, more blatantly about the last night of Jesus.
I think the Hooters considered themselves a Christian band, so I’m pretty sure “All You Zombies” doesn’t really qualify as a non-religious song.
Again, since the entire song is nothing but that one Corinthians verse, I don’t know if you could call it non-religious. If Death Cab for Cutie did a song which started “In the beginning the world was without form and void” and sang nothing but Genesis 1-3 with a syncopated upbeat, I think it would still qualify as a religious song.
Well, depending on whether you consider him secular or not, probably nearly half of Dylan’s songs have some kind of passing Bible reference.
For instance “Shelter from the Storm” includes “In a little hilltop village / they gambled for my clothes” which I’m pretty sure comes from the New Testament story of Roman soldiers playing dice to see who got Jesus’ cloak. You can find something like that in many, many, many Dylan songs, just little fragments, but a real story is referenced.
In fact, I think this list could get very, very long. The Bible is a pretty big cultural reference, even for godless rockers. Here, I’ll add “Sympathy for the Devil”, where the narrator is around for Jesus’ “moment of doubt and pain”.
Pearl Jam’s Given To Fly has Christ-like references: