Washington Redskins, every home game since 1968. The Green Bay Packers have sold out every game at Lambeau Field since 1960. However, the Packers played a portion of their home schedule at Milwaukee's County Stadium until 1993. There were some games during the 1970s that did not sell out in Milwaukee.
They have sold out every game since 1960. So say the Redskins have more. But they have only sold out every game since 1968. So if my math is correct Green Bay has more.
Currently the Denver Broncos have sold out every home game they have ever played including games prior to their induction into the NFL, when those records were not kept.
Otto Graham of the Cleveland Browns started 6 consecutive NFL Championship Games between 1950-1955.
New England Patriots