"I'm A Believer", first made famous by The Monkees (Neil Diamond recorded his version a few months after The Monkees), and then much later, brought back into the mainstream by Smash Mouth.
"That Was Then, This Is Now" brought the Monkees back to the Top 40 during their '80s revival. It was originally written and recorded by a now-obscure '80s pop band called The Mosquitos.
Several of the Tommy Boyce/Bobby Hart songs were recorded by other groups first, such as "(I'm Not Your) Steppin' Stone" (Paul Revere & The Raiders) and "Words" (The Leaves).
Narm: "The Day We Fall in Love" from More of the Monkees would otherwise be your typical schmaltzy Davy Jones ballad, but in this one, Davy speaks the lyrics instead of singing them.
Vindicated by History: Despite the "manufactured band" thing, the band's music has held up quite well. You even had the Sex Pistols (another manufactured band) and Minor Threat covering "(I'm Not Your) Steppin' Stone".