Teams Losing a Finals Game 7 at Home

It's actually a lot more common than you might think. Much more so in baseball (20 times) than in basketball (4) and hockey (5). The Yankees have done it 3 times, but have caused it 4 times.

1909 World Series: Detroit Tigers at Bennett Park, to Pittsburgh Pirates
1926 World Series: New York Yankees at Yankee Stadium, to St. Louis Cardinals
1934 World Series: Detroit Tigers at Navin Field (Tiger Stadium), to St. Louis Cardinals
1945 Stanley Cup Finals: Detroit Red Wings at Olympia Stadium, to Toronto Maple Leafs
1945 World Series: Chicago Cubs at Wrigley Field, to Detroit Tigers
1952 World Series: Brooklyn Dodgers at Ebbets Field, to New York Yankees
1955 World Series: New York Yankees at Yankee Stadium, to Brooklyn Dodgers
1956 World Series: Brooklyn Dodgers at Ebbets Field, to New York Yankees
1957 World Series: New York Yankees at Yankee Stadium, to Milwaukee Braves
1958 World Series: Milwaukee Braves at Milwaukee County Stadium, to New York Yankees
1962 World Series: San Francisco Giants at Candlestick Park, to New York Yankees
1965 World Series; Minnesota Twins at Metropolitan Stadium, to Los Angeles Dodgers
1967 World Series: Boston Red Sox at Fenway Park, to St. Louis Cardinals
1968 World Series: St. Louis Cardinals at Busch Memorial Stadium, to Detroit Tigers
1969 NBA Finals: Los Angeles Lakers at The Forum, to Boston Celtics
1971 Stanley Cup Finals: Chicago Blackhawks at Chicago Stadium, to Montreal Canadiens
1971 World Series: Baltimore Orioles at Memorial Stadium, to Pittsburgh Pirates
1972 World Series: Cincinnati Reds at Riverfront Stadium, to Oakland Athletics
1974 NBA Finals: Milwaukee Bucks at Milwaukee Arena (MECCA), to Boston Celtics
1975 World Series: Boston Red Sox at Fenway Park, to Cincinnati Reds
1978 NBA Finals: Seattle SuperSonics at Seattle Center Coliseum, to Washington Bullets
1979 World Series: Baltimore Orioles at Memorial Stadium, to Pittsburgh Pirates
2009 Stanley Cup Finals: Detroit Red Wings at Joe Louis Arena, to Pittsburgh Penguins
2011 Stanley Cup Finals: Vancouver Canucks at Rogers Arena, to Boston Bruins
2014 World Series: Kansas City Royals at Kauffman Stadium, to San Francisco Giants
2016 NBA Finals: Golden State Warriors at Oracle Arena, to Cleveland Cavaliers
2016 World Series: Cleveland Indians at Progressive Field, to Chicago Cubs
2017 World Series: Los Angeles Dodgers at Dodger Stadium, to Houston Astros
2019 Stanley Cup Finals: Boston Bruins at TD Garden, to St. Louis Blues

The pre-Super Bowl NFL Championship Game, 1933-1965, not put at a neutral site, was a clinching game, but not a Game 7. Nevertheless, there were 9 home-team losers, and the Giants were 4 of them:

1937 Chicago Bears at Wrigley Field, to Washington Redskins
1940 Washington Redskins at Griffith Stadium, to Chicago Bears
1944 New York Giants at Polo Grounds, to Green Bay Packers
1946 New York Giants at Polo Grounds, to Chicago Bears
1949 Los Angeles Rams at Los Angeles Coliseum, to Philadelphia Eagles
1952 Cleveland Browns at Cleveland Municipal Stadium, to Detroit Lions
1955 Los Angeles Rams at Los Angeles Coliseum, to Cleveland Brown
1958 New York Giants at Yankee Stadium, to Baltimore Colts
1962 New York Giants at Yankee Stadium, to Green Bay Packers