Ok, here's what I have so far (thanks also to people at another forum for helping me out).
1) Pumping Iron for Enya ~ can't remember his name...
2) Natalie Portman ~ Ozma
3) Song for Lennon ~ Queen
4) You're So Vain ~ Carly Simon? (Warren Beatty?)
5) Do It With Madonna ~ some aussi band
6) Clint Eastwood ~ Gorillaz
7) Candle in the Wind ~ Elton John (Monroe)
8) Shut Your Mouth ~ Garbage (Madonna)
9) Red Vines ~ Aimee Mann (Paul Thomas Anderson)
10) Love Ridden ~ Fiona Apple (ditto)
11) Dreaming of the Queen ~ Pet Shop Boys (Diana?)
12) Thief of Hears ~ Madonna (Benning?)
13) Everybody Hurts ~ REM (Phoenix)
14) Ballad of Chasey Lane ~ Bloodhound Gang
15) River ~ Natalie Merchant (Phoenix)
16) David Duchovny ~ Bree Sharp
17) Robert DeNiro's Waiting ~ Bananarama
18) Bette Davis Eyes ~ Kim Carnes
19) Abraham, Martin and John ~ Dion
20) Meet James Ensor ~ TMBG
21) James K. Polk ~ TMBG
22) Memphis ~ PJ Harvey (Jeff Buckley)
23) Mr. Crowley ~ Ozzy (Alistair Crowley)
24) Yeats's Grave ~ The Cranberries (W.B. Yeats)
25) Mulder and Scully ~ Catatonia
26) Rivers ~ Sugar Ray (Rivers Cuomo)
27) Ringo, I Love You ~ Stereo Total (Ringo Starr) [not the original performer though]
28) How Do You Sleep At Night? ~ John Lennon (Paul McCartney)
29) Alan Cumming ~ Turd Ferguson
30) Cherry Lips ~ Garbage (JT LeRoy)
31) Hats Off To Roy Harper ~ Led Zep
32) Courtney Love ~ Nerf Herder
33) Oh Yoko ~ Lennon (Yoko Ono)
34) Barry White ~ Fun Loving Criminals
35) Buddy Holly ~ Weezer
Phew!! That's a lot already! I might have to do some serious uhm... cutting out songs such as things that aren't really about the people but for now the list looks great! If you still remember something new, don't hesitate to tell me. Hell, I could make two tapes out of this material!
EDIT: Corrected some things in the list.