Records of teams that player in the HoF game going back to the 2000 season:
2000: New England Patriots (5-11, NP) San Fransisco 49'ers (6-10, NP)
2001: St. Louis Rams (12-4, P) Miami Dolphins (11-5, P)
2002: New York Giants (10-6, P) Houston Texans (4-12, NP)
2003: Kansas City Chiefs (13-3, P) Green Bay Packers (10-6, P)
2004: Washington Redskins (6-10, NP) Denver Broncos (10-6, NP)
2005: Chicago Bears (11-5, P) Miami Dolphins (9-7, NP)
2006: Oakland Raiders (2-14, NP) Philadelphia Eagles (10-6, P)
2007: Pittsburgh Steelers (10-6, P) New Orleans Saints (7-9, NP)
2008: Washington Redskins (8-8, NP) Indianapolis Colts (12-4, P)
Not too much you can derive from that, but if someone were inclined, you could dig deeper and see how those teams started out relative to their division opponents perhaps.
My guess though, is that the extra game is a small factor in the success or failure of a franchise in any given season.