The music that plays when Mario/Luigi loses a life is basically the "E-I-E-I-O" portion of "Old MacDonald Had A Farm".
The first ground theme is basically a speeded-up interpretation of Sentimental Journey (especially the Ringo Starr version).
But then again, "Bringer of War" is a pretty appropriate theme for those levels. The similarities are even more obvious in the orchestrated version from Super Mario Galaxy.
The Airship music is suspiciously similar to of Mars: The Bringer Of War from Holst's "The Planets" suite.
This hook also appears as a Recurring Riff in the "Snowman's Land" music in Super Mario 64.
The second ground theme heavily resembles the The Buggles' "Video Killed The Radio Star", especially the second half.
In the Super Mario All-Stars remakes of the first game and The Lost Levels, the final Bowser battle theme is similar to the Macbeth theme from Star Fox.
The underwater theme is very similar to the Waltz 5B section of Johann Strauss's "Blue Danube", and to a lesser extent Tchaikovsky's "Waltz of the Flowers" from The Nutcracker.
The first six notes of the original theme is actually suspiciously similar to the Minsky Pickup.
The castle theme sounds like Nikolai Rimsky-Korsakov's "Flight of the Bumblebee".