The Mount Union Purple Raiders have 55 consecutive victories, the most of any division of NCAA football. But they were not in "consecutive weeks". In addition to having one or more "bye weeks" during the regular season, there is a 8-9-month period between seasons to break up the "consecutive weeks". In fact, if you keep this "consecutive weeks" as part of the question, it's going to be very difficult for anyone to answer this question, and I don't see why the answer would be meaningful to anyone.
SEC
The most consecutive losses in NCAA football in Division 1 (FBS) belongs to the Northwestern Wildcats. They lost a total of 34 games in a row between 1979 and 1982.
Penn State
Michigan has the most wins in Division I-A college football. From 1907 to 1917, Washington football teams were unbeaten in 63 consecutive games, fielding 1930 points, and allowing only 118, still an NCAA Division I-A record
University of Miami,Florida. 58 consecutive wins,an NCAA Record
The University of Oklahoma
Boise State
Nebraska
Wilmington college
USC
Northwestern wildcats - 30 1980