Continuity mistake: When Charlie Allnut grabs a bottle of gin from the box, all the bottles in it are wrapped with paper for transport. But when Rose Sayer takes the bottles out of the box to empty them into the river, the paper is gone (from the movements of her hand/wrist, you can tell that she doesn't unwrap the paper from the bottles).
Continuity mistake: In the scene where Humphrey Bogart and Katharine Hepburn are taking a bath in the river, Katharine starts off looking as if she is bathing fully undressed. As she climbs into the boat, however, she is clearly seen with a full set of underclothing that includes shoulder straps which were not there earlier.
Continuity mistake: When Rose and Charlie are getting married on the Louisa, one camera view of the couple shows a small island in the background, about 50 yards off the port side. The island stays in the exact same place in all subsequent shots from that angle, indicating that the ship is stationary. When they cut to shots looking towards the bow of the Louisa bearing down on the capsized African Queen, it is traveling at a good pace, as evidenced by the bow wake.