As far as Basketball-Reference can look it up, Ryan Bowen had a streak of 43 games without a turnover from December 9, 2004 to April 1, 2005:
http://www.basketball-reference.com/pla ... i?id=Ddx3yI'm working also on most games with at least 1 assist, because guys like Ryan Bowen might get a lot of no-turnover games because they don't play or handle the ball much.
EDIT: For games with at least 1 assist and no turnovers, the record is 11, held by Steve Kerr (three separate occasions) and Reggie Miller.