Born in Orange, New Jersey, into a baseball family, Feeney was the grandson of Charles Stoneham, principal owner of the New York Giants until his death in 1936, and the nephew of Horace Stoneham, who owned the team from 1936 through 1976. Feeney began his association with the Giants as a batboy, and after his graduation from Dartmouth College and military service during World War II he joined the team's front office at the age of 24 as vice president in 1946. Although he never held the official title of general manager, Feeney would function as head of the Giants' baseball operations department for almost 24 years.
In 1951, the Giants battled back from a 13½ game deficit on August 11, winning 37 of their last 44 games to force a best-of-three pennant playoff with Brooklyn. After splitting the first two games, the Giants overcame one last hurdle — a 4-1, ninth-inning Brooklyn lead in Game 3 — to beat the Dodgers on Bobby Thomson's three-run home run, baseball's version of the "Shot Heard 'Round the World." It was New York's first National League pennant since 1937, but the Giants dropped the 1951 World Series in six games to the New York Yankees.
Brooklyn dominated the NL for the next two seasons, but, in 1954, Durocher's Giants — led by batting champion Willie Mays and the runner-up, Don Mueller — emerged as champions, winning the pennant by five games. Drawing the Cleveland Indians (who had set an American League record by winning 111 games) as their opponents in the 1954 World Series, the Giants won in four straight games, highlighted by Mays' game-saving catch of Vic Wertz' long drive in Game 1, the clutch hitting of obscure outfielder and pinch hitter Lamar "Dusty" Rhodes, and effective pitching from four different starters.
Unfortunately, the 1954 Fall Classic was the last highlight of the Giants' 70-plus year history in New York. Attendance plunged in the years that immediately followed, and after Durocher's resignation in 1955 to become a "Game of the Week" baseball broadcaster, the team played poorly. By 1957, owner Stoneham had decided to leave for greener pastures, ultimately choosing San Francisco as the team's destination to preserve its historic rivalry with the Dodgers, who simultaneously moved to Los Angeles.
In 1962, the Giants and Dodgers engaged in a West Coast-version of the 1951 pennant chase. Los Angeles built an early lead, but began to fall to earth when ace left-hander Sandy Koufax was sidelined for the season by a finger ailment. By season's end, the teams were deadlocked, at 101 wins and 61 defeats. Again, a best-of-three playoff would determine the champion, and — again — the Giants would rally in the ninth inning of Game 3 to beat the Dodgers. But the deciding game was played in Los Angeles, thus the winning run — forced in by a bases-loaded walk — was not a "walk-off" situation and lacked the drama of Thomson's home run. The Giants, as in '51, faced the Yankees in the 1962 World Series and lost, this time in seven games.
Although San Francisco remained a first-division team, and frequent contender, during the rest of the 1960s, it did not win another pennant during the decade. It finished in second place for four successive seasons. By 1969, the team was showing signs of age and decline. Concurrently, Feeney was being considered for prominent positions within Major League Baseball's hierarchy. After his candidacy for Commissioner of Baseball fell short, Feeney succeeded Warren Giles as NL president late in 1969.
Just before his tenure as NL president ended, he made an appearance on Jeopardy! in the show's second season in 1986.
As he passed his 65th birthday, Feeney was succeeded as NL president by A. Bartlett Giamatti. His baseball career concluded with a 15-month tour as president of the San Diego Padres (1987-88), which ended with his resignation the day after he made an obscene gesture to hecklers on Fan Appreciation Night in San Diego on September 24, 1988.
He died in January 1994 of a heart attack in San Francisco at the age of 72.