Ooh, this thread finally happened. Let's see...
Sonic 1 has a bunch of these to varying degrees:
The ascending bassline in
Bobby Brown's Every Little Step shows up at the start of
Spring Yard Zone. It's debatable, but there's enough stylistic similarity between the two songs that I'm willing to argue it's the source.
Andy Williams' Music To Watch Girls By has obvious similarities to
Marble Zone.
The first half of the synth line running throughout
Duran Duran's Planet Earth is a close match for the opening bars of
Final Zone.
Blade Runner (End Titles) by Vangelis is so clearly the source for most of
Scrap Brain Zone that it hurts. Same basic chord progression and similar instrumentation all the way down to the occasional timpani fill.
There's also an easy one in 8-bit Sonic 2: Compare the
boss theme with
808 State's Cubik.
Going the other way, most people know that Sonic 2's ending became
Sweet Sweet Sweet, but Nakamura also turned Green Hill Zone into
Marry Me and Starlight Zone into
Kusuriyubi no Kesshin. At least he owns the source material in all three cases.
Sonic CD and Sonic Rush also borrow samples from all over the place, but that's a post for another time (specifically, when someone's managed to get an upload of the original version of Public Enemy's
Bring the Noise to stick on Youtube for more than five seconds).
Moving over to a different franchise, Mega Man X has its own fair share.
Armored Armadillo pulls its opening from the
Peter Gunn theme, and the developer's love of Guns N' Roses that came to a head in X5 cropped up earlier in the series with X3's
Neon Tiger borrowing riffs from
My Michelle. I'm sure there are others, but I'll leave that to someone with more familiarity with the series.
Also, Bobby Prince's musical larceny regarding DOOM is widely documented, but he also composed the music for a bunch of other games. Ignoring the public domain, what is
easily the best example of him knocking off someone else's work is the
main theme of Cosmo's Cosmic Adventure, which is little more than a mildly modified version of
ZZ Top's Tush. I wish I could find a full recording of it, it keeps going on well past what's in that video. If the similarity itself doesn't convince you, the actual audio file
is named ZZTop. There's also one named
Devo, though I couldn't begin to tell you what of their discography it's based on, if anything.