As a sort of honorable mention to this, in 1971 Padres SS Enzo Hernandez was allowed 549 official at bats (by Preston Gomez), and yet somehow managed to drive in only 12 runs. Twelve.
The LinkA pretty remarkable feat by itself, and during that season he had separate streaks of 32, 24, and 21 games without an RBI. In 1971, Enzo Hernandez was a non-RBI machine.
(Luis Castillo, then of FLA, challenged Hernandez' record in 2000, but screwed it up at the end, and ended up with 17 RBIs in 539 ABs. No real long streaks in there, though; Castillo spaced his RBIs out evenly. He did manage to go the first 18 games of the season without driving in a run, whihc may be a record for an everyday player, I don't know.)