No, it didn't.
2016: Obama's America was released on Aug. 24, 2012.
On Aug. 21, 2012, Obama's approval rating (as per Rasmussen) was 49%. On Aug. 27, it was 50%. It would hover between 48-50 throughout the month of September.
Daily Presidential Tracking Poll - Rasmussen Reports™
His approval as per Gallup was 47% the week before the "documentary" was released; it remained within a point either way through September.
His RCP poll average (the average of all the polls) had him around 48% at the time of the release, and between 47 and 49 percent for the entire month of September (except for a brief flirtation with 50%).
Any effect on Obama's polling caused by that "documentary" was statistically insignificant to the point of nonexistent. Or, in other words, you simply made that up.