Here's the list, which I'll edit every few songs or so:
40. Rock On - Michael Damian (1 week, 1989)
39. Good Thing - Fine Young Cannibals (1 week, 1989)
38. Sledgehammer - Peter Gabriel (1 week, 1986)
37. Invisible Touch - Genesis (1 week, 1986)
36. Dirty Diana - Michael Jackson (1 week, 1988)
35. Live To Tell - Madonna (1 week, 1986)
34. Baby Don't Forget My Number - Milli Vanilli (1 week, 1989)
33. Shakedown - Bob Seger (1 week, 1987)
32. Holding Back The Years - Simply Red (1 week, 1986)
31. Beatles Medley - Stars On 45 (1 week, 1981)
30. You Keep Me Hanging On - Kim Wilde (1 week, 1987)
29. Higher Love - Steve Winwood (1 week, 1986)
28. Everytime You Go Away - Paul Young (1 week, 1985)
27. I Still Haven't Found What I'm Looking For - U2 (2 weeks, 1987)
26. Everybody Wants To Rule The World - Tears For Fears (2 weeks, 1985)
25. Jessie's Girl - Rick Springfield (2 weeks, 1981)
24. Monkey - George Michael (2 weeks, 1988)
23. Toy Soldiers - Martika (2 weeks, 1989)
22. A View To A Kill - Duran Duran (2 weeks, 1985)
21. Papa Don't Preach - Madonna (2 weeks, 1986)
20. The Power Of Love - Huey Lewis and the News (2 weeks, 1985)
19. It's Still Rock and Roll To Me - Billy Joel (2 weeks, 1980)
18. I Wanna Dance With Somebody - Whitney Houston (2 weeks, 1987)
17. The Reflex - Duran Duran (2 weeks, 1984)
16. The Flame - Cheap Trick (2 weeks, 1988)
15. Heaven - Bryan Adams (2 weeks, 1985)
14. Alone - Heart (3 weeks, 1987)
13. Don't You Want Me - The Human League (3 weeks, 1982)
12. On My Own - Patti LaBelle and Michael McDonald (3 weeks, 1986)
11. La Bamba - Los Lobos (3 weeks, 1987)
10. Right Here Waiting - Richard Marx (3 weeks, 1989)
9. Coming Up - Paul McCartney (3 weeks, 1980)
8. Ghostbusters - Ray Parker Jr. (3 weeks, 1984)
7. Shout - Tears For Fears (3 weeks, 1985)
6. Roll With It - Steve Winwood (4 weeks, 1988)
5. Magic - Olivia Newton-John (4 weeks, 1980)
4. When Doves Cry - Prince (4 weeks, 1984)
3. Eye Of the Tiger - Survivor (6 weeks, 1982)
2. Every Breath You Take - The Police (8 weeks, 1983)
1. Endless Love - Lionel Richie and Diana Ross (9 weeks, 1981)
Oddly, "Time After Time" was omitted though that song also spent 2 weeks at #1.