Don't get me wrong -- 24 quarters without an offensive touchdown is pitiful, whether you have a fourth-string quarterback or a street-clothes tryout starting for your team. However, the Cleveland Browns did not break the NFL record in terms of an offensive  touchdown scoring drought:

The Browns' 31-0 loss to the Steelers was the sixth consecutive game in which Cleveland failed to score an offensive touchdown-that is, a TD by rushing or passing. That's the longest such streak at any point in any season since the Eagles posted a six-game streak 72 years ago.

To clear up some misinformation on the Browns-Steelers telecast, Cleveland's streak of 24 consecutive quarters without an offensive touchdown was not an NFL record-not even close, in fact. The single-season record is 32 consecutive quarters without an offensive TD.

Link

Even if the Browns go without an offensive touchdown for 2+ games next season, we won't break the record because the record is labeled for a single season. I'm not sure if a longer record exists over multiple seasons.