I added songs from older and more modern films to the list.
The Magnificent Seven - 49 Hawaii Five-O - 65 Valley of the Dolls - 67 Shaft - 72 Mission: Impossible - 75 Gone With the Wind (Tara's Theme) - 76 Zorba the Greek - 78 Jean [The Prime of Miss Jean Brodie] - 83 The Good, the Bad and the Ugly - 89 For All We Know [Lovers and Other Strangers] - 94 The Green Leaves of Summer [The Alamo] - 98 M*A*S*H (Suicide Is Painless) - 98 Nobody Does It Better [The Spy Who Loved Me] - 106 Viva Las Vegas - 107 Hi-Lili, Hi-Lo - 111 Star Wars - 113 Chim Chim Cheree [Mary Poppins] - 115 Tammy - 116 Born Free - 116 You Only Live Twice - 117 Love Theme from "A Star Is Born" (Evergreen) - 117 Three Coins In the Fountain - 118 (Everything I Do) I Do It for You [Robin Hood: Prince of Thieves] - 118 Pink Panther Theme - 122 Against All Odds (Take a Look at Me Now) - 125 A Whole New World [Aladdin] - 127 Lullaby of Broadway [Gold Diggers of 1935] - 133 From Russia With Love - 138 I Just Called to Say I Love You [The Woman In Red] - 139 Around the World (In 80 Days) - 140 High Noon (Do Not Forsake Me) - 142 Let It Go [Frozen] - 143 Peter Gunn - 144 Circle of Life [The Lion King] - 145 Live and Let Die - 147 James Bond Theme - 150 Two for the Road - 158 Chattanooga Choo Choo [Sun Valley Serenade] - 175 Exodus (+ This Land Is Mine) - 178 You'll Never Know [Hello, Frisco, Hello] - 180 Jeepers Creepers [Going Places] - 181 Beauty and the Beast - 181 Can You Feel the Love Tonight? [The Lion King] - 182 Theme from "Romeo and Juliet" (+ A Time for Us) - 187 Mrs. Robinson [The Graduate] - 189 Charade - 203 My Heart Will Go On [Titanic] - 204 Whatever Will Be, Will Be (Que sera, sera) [The Man Who Knew Too Much] - 207 Goldfinger - 220 Long Ago (and Far Away) [Cover Girl] - 224 True Love [High Society] - 224 Never On Sunday - 230 The Way We Were - 238 Blues In the Night - 244 Emily [The Americanization of Emily] - 248 Summer of '42 - 251 Love Theme from "The Godfather" - 257 Help! - 267 I Will Wait for You [The Umbrellas of Cherbourg] - 274 Mona Lisa [Captain Carey, U.S.A.] - 275 All the Way [The Joker is Wild] - 277 Love Is a Many-Splendored Thing - 288 Love Letters - 307 What Are You Doing the Rest of Your Life? [The Happy Ending] - 312 More [Mondo Cane] - 314 That Old Black Magic [Star Spangled Rhythm] - 315 Love Story - 321 A Hard Day's Night - 326 Lara's Theme (+ Somewhere My Love) [Doctor Zhivago] - 326 Alfie - 331 Pennies From Heaven - 332 The Windmills of Your Mind [The Thomas Crown Affair] - 340 Raindrops Keep Fallin' on My Head [Butch Cassidy and the Sundance Kid] - 346 Baby, It's Cold Outside [Neptune's Daughter] - 425 Cheek to Cheek [Top Hat] - 430 It Might As Well Be Spring [State Fair] - 433 The Look of Love [Casino Royale] - 476 I've Got You Under My Skin [Born To Dance] - 478 Let It Be - 491 The Days of Wine and Roses - 498 When You Wish Upon a Star [Pinocchio] - 500 Unchained Melody - 512 They Can't Take That Away from Me [Shall We Dance] - 526 The Shadow of Your Smile [The Sandpiper] - 613 Silver Bells [The Lemon Drop Kid] - 746 Moon River [Breakfast at Tiffany's] - 795 Over the Rainbow [The Wizard of Oz] - 1283 Have Yourself a Merry Little Christmas [Meet Me In St. Louis] - 1652 White Christmas [Holiday Inn] - 1814
The biggest surprise to me on the above list was the popularity of 1945's "It Might As Well Be Spring," which is being covered today as much as it was when it was first written.
The winning composer is Irving Berlin with 2,244 covers of his tunes.
Runners up:
Henry Mancini: 2,036 Hugh Martin and Ralph Blane: 1,652 Harold Arlen and Yip Harburg - 1,283 The Beatles: 1,084 Michel Legrand: 926 John Barry: 863 Johnny Mandel: 861
It would be interesting to see Bob's list with all songs removed (either those from musicals or themes that had a popular vocal version to enable greater propagation).