In Colorado's 1-0 win over the Giants tonight, there were no RBI in the game. The winning run was credited as scoring on an error by San Francisco catcher Bengie Molina while trying to tag Garret Atkins on a play at home.
Back on August 11, 1971, the Reds scored five times without an RBI thanks to errors by Jack Billingham, Bob Watson, Joe Morgan, and Denis Menke of the Astros.
And on July 1, 1990, the White Sox scored four runs without an RBI. Or even a hit. And Andy Hawkins took the loss for the Yankees.
Four runs without an RBI for either team appears to be the high water mark in Baseball-Reference.com's Play Index. I would not be surprised if the 19th Century yielded a higher total since there were so many more errors then.