It depends how closely you want to define "home."
If we say that a home Super Bowl would be one that takes place in the same metro area as a team, then two teams have had a home Super Bowl.Super Bowl XIV (1979-80) at the Rose Bowl in Pasadena, California saw the L.A. Rams lose to Pittsburgh. Super Bowl XIX (1984-85) was at Stanford Stadium in the San Francisco Bay Area, and in it San Francisco beat Miami.
If you require a home Super Bowl to be one that takes place at the home stadium of the team, then Miami and Houston have gotten the closest: they have each lost in the playoffs only two games away from a home Super Bowl.
A team entering the playoffs in a year that its stadium was hosting the Super Bowl has happened 7 times:Super Bowl V (1970-71) in Miami: Dolphins lost first round. (2 games away) Super Bowl XIII (1978-79) in Miami: Dolphins lost first round. (3 games away) Super Bowl XXIX (1994-95) in Miami: Dolphins lost second round (2 games away)...