The Knickerbockers’ “Lies” sounds like the Beatles
Klaatu’s “Sub Rosa Subway” (among others) sounds like the Beatles
Mouse and the Traps’ “Public Execution” sounds like Bob Dylan
Fontella Bass’s “Rescue Me” sounds like Aretha Franklin
The Jags’ “Back of My Hand (I Got Your Number)” sounds like Elvis Costello
Divine Sounds’ “What People Do For Money” sounds like Run DMC
Ali Thompson’s “Take a Little Rhythm” sounds like Paul McCartney
Eddie Holland’s “Jamie” sounds like Jackie Wilson
Junior’s “Mama Used to Say” sounds like Stevie Wonder
John Cafferty’s “On the Dark Side” sounds like Bruce Springsteen
Oddly enough, the Rutles sound a lot like the Beatles (or is it the other way around?) 
America’s “A Horse With No Name” sounds so much like Neil Young it’s scarey.
Rush’s “Working Man” sounds like Tony Iommi (from Black Sabbath) sat in on the sessions
Bonnie Tyler sounds exactly like Rod Stewart. Occasionally, so does Melissa Etheridge
ELO often sounds like The Beatles
I’m sure there are still many people today who think that The Hollies’ Long Cool Woman (In a Black Dress) is by Creedence Clearwater Revival.
Most of 80s r&b band Ready For The World’s tunes sound like reworked Prince and the Revolution, whom the band idolized.
Sometimes, The Violent Femmes sounded quite a bit like The Velvet Underground. (Especially at the start.)
Mott The Hoople’s All The Young Dudes sounds like Bowie is singing it, which is weird, 'cause Bowie wrote it.
Almost everything in the 80’s sounded like Prince, probably because Prince wrote almost all of it!
Tom Petty and the Heartbreakers’ “American Girl” sounds like the Byrds.
And the band Dirty Looks sounds a LOT like AC/DC. (At least back in the late 80’s/early 90’s – haven’t heard anything new by them in a long while.)
All the songs by Jet sound like The Black Crowes.
The first time I heard “Sultans of Swing” by Dire Straits I thought it was Dylan.
I forgot one – on the latest Billy Idol CD there is a song that sounds exactly like Neil Diamond. “Cherie” or something like that. I wish I was kidding. And I wish I was kidding when I said I liked it.
That reminds me of what ought to be the #1 example in this thread:
Stuck in the Middle With You, by Stealer’s Wheel.
I think most people think it is Dylan. (I know some of you are here right now. How else could this not have been mentioned already? :D)
“Spill the Wine” by Eric Burdon sounds like a Rolling Stones song to me. Not a good Rolling Stones song, but nonetheless…
I thought Coldplay was U2 for a couple months when they first got a lot of air time…
At The Drive-In reminds me even more of Rush.
The lead singer of The Dirty Pictures sounds incredibly like the lead singer of Dada. And the lead singer of Prime Sth sounds much like the lead singer for Collapsis.
I can’t count the number of people, me included, who thought “Why I’m Here” by Oleander was a previously unreleased Nirvana song.
I’m pretty sure “Your Cover’s Blown” is the first and only Belle and Sebastian song I can stand because it reminds me of “The Way You Walk” by Papas Fritas.
One more. I haven’t heard anything else by these bands per se, but Just Like Hemingway by Blue Clocks Green actually did made me think it was a Smiths song, as did Sound of Settling by Death Cab for Cutie.
Bowling for Soup sounds like Weird Al Yankovic.
Elvis’s “Blue Suede Shoes” sounds like Carl Perkins’s “Blue Suede Shoes.” Does that count?