The Indianapolis Colts
The New England Patriots
26 consecutive losses by Tampa Bay from 1976-77 http://www.football.com/nfl/records/NFLTGamesLost.shtml
Tampa Bay Buccaneers with 26 consecutive losses in 1976-1977.
Dallas Cowboys - 1975-1983
As of the start of the 2008 season, the St. Louis Rams and the Minnesota Vikings have the most franchise playoff losses with 24. The Vikings playoff record is 18-24 and the Rams playoff record is 19-24.