“Beautiful”–also Christina Aguilera and Snoop Dog
“Magic” – also Pilot and the Cars
“Forever Young” – also Bob Dylan
“Do It Again” – also Steely Dan
“Help Me” – also Sonny Boy Williamson
And Jefferson Starship
And Olivia Newton-John. And the Cars. 
A couple more that came to mind…
“On the Road Again” Bob Dylan/Willie Nelson
“Hold On” Ian Gomm/Triumph/Wilson Philips/En Vogue
“Where Is The Love” Roberta Flack and Donny Hathaway/Black Eyed Peas featuring Justin Timberlake
“S.O.S” ABBA/Rhianna
“Runaway” Del Shannon/Jefferson Starship/Bon Jovi/Roxanne Shante
Also Soundgarden
Also the Misfits
Also Steely Dan and Queens of the Stone Age
Still more…
“Words” The Monkees/The Bee Gees
“It’s My Life” The Animals/Talk Talk [covered by No Doubt]
“Talk Talk” The Music Machine/Talk Talk
“Hurt” Timi Yuro [covered by Elvis Presley]/Nine Inch Nails [covered by Johnny Cash]
“Separate Ways” Elvis Presley/Journey
“Woman” Peter and Gordon/John Lennon
“Bitch” The Rolling Stones/Meredith Brooks
“Dreams” Fleetwood Mac/Van Halen
“Heartbreak Hotel” Elvis Presley/Whitney Houston
“Express Yourself” Charles Wright & the Watts 103rd Street Band/Madonna
“Scorpio” Dennis Coffey/Grandmaster Flash and the Furious Five
“America” The cast of ‘West Side Story’/Simon and Garfunkel
(I just know I’m not getting any work done this afternoon.)
“Do Re Mi” - Woody Guthrie, Rodgers & Hammerstein
And Seal.
…And Eurythmics.
“Cherish” The Association/Madonna
“Human Nature” Michael Jackson/Madonna
“Over and Over” The Dave Clark Five/Madonna/Nelly & Tim McGraw
“Deeper and Deeper” Freda Payne/Madonna
“The Power of Love” Joe Simon/Huey Lewis and the News/Jennifer Rush [covered by Celine Dion]
“Baby I Love You” The Ronnettes/Aretha Franklin
“Summertime” George & Ira Gershwin/DJ Jazzy Jeff & the Fresh Prince
“Crazy Love” Van Morrison/Poco
“Suspicion” Terry Stafford/REM
And Yes.
Ashes to Ashes" by David Bowie and the Vinnie Vincent Invasion
Some additions:
[ul]
[li]“I Need You”, add America[/li][li]“Jump”, add the Pointer Sisters. It even came out later in the same year as Van Halen’s song.[/li][li]“The One”, add Elton John[/li][li]“Jane”, add Jefferson Starship[/li][/ul]
And before anybody mentions it, Peter Schilling’s “Major Tom” pairing with David Bowie’s song doesn’t count because David Bowie’s version is called “Space Oddity”.
And Sarah Maclachlan.
“Fire” Jimi Hendrix/The Crazy World of Arthur Brown/The Ohio Players/The Pointer Sisters
“War” Edwin Starr/Bob Marley
“Again” Mel Torme/Lenny Kravitz
“Let’s Dance” Chris Montez/David Bowie
“Poison” Alice Cooper/Bel Biv Devoe
“Heartbreaker” Pat Benatar/Dionne Warwick
“Don’t Be Cruel” Elvis Presley/Bobby Brown
“All Around The World” Little Richard/The Jam
“Candy” Johnny Mercer/Iggy Pop/Mandy Moore
“Rock the Boat” The Hues Corporation/Aaliyah
“Miss You” The Rolling Stones/Aaliyah
“I’m Free” The Rolling Stones/The Who
“Family Affair” Sly and the Family Stone/Mary J. Blige
“Me, Myself and I” De La Soul/Beyonce
“Heaven” Talking Heads/Bryan Adams/Warrant/Los Lonely Boys
“My Love” Petulia Clark/Paul McCartney/Justin Timberlake
“Sara” Fleetwood Mac/Starship
“Let Me Go” Heaven 17/Three Doors Down
“Because of You” Tony Bennett/Kelly Clarkson
“Learning to Fly” Tom Petty and the Heartbreakers/Pink Floyd
“The Show Must Go On” Pink Floyd/Queen
“Somebody to Love” Jefferson Airplane/Queen
Bad Reputation by Joan Jett, Thin Lizzy, and Sammy Hagar
“Someday”–Mariah Carey, The Strokes, piece-of-shit Sugar Ray
“Right Here, Right Now”–Jesus Jones, Fatboy Slim
The one you list as Andy Gibb’s was actually recorded by his brothers Barry, Robin, and Maurice, a.k.a. the Bee Gees.
“Wild Thing” – The Troggs/Tone-Loc
“Don’t Stop” – Fleetwood Mac/The Rolling Stones
“Heartbreaker”
Led Zepplin
Grand Funk
Nantucket
Pat Benatar
Can’t say I am familiar with the Dionne Warwick version
Kool & the Gang also did a (terrible) song called “Cherish,” and Luther Vandross had a “The Power of Love.”
Doo doo doo doo doo not forget the Rolling Stones.