Other mistake: About 3/4 of the way through the film a boat arrives at the left quay of the Seine. Looking west down the river with Notre Dame on the right of the scene you can see a tour bus cross the bridge.
Plot hole: The impetus of the plot is that the doctor mismatches the two sets of twins. But watch carefully the placement as they are born: first the DiCici baby, then the Coupe on the left, then the other DiCici, then the other Coupe on the right of the other three, making it, from the viewer's perspective, DiCici-Coupe-DiCici-Coupe. The doctor then switches the two middle ones, which supposedly mix the twins, but in fact it makes them DiCici-DiCici-Coupe-Coupe, and the two left and two right are given to the fathers. So the twins are not mismatched.
Continuity mistake: When the two different brothers retrieve the book from the Narrator (and ultimately shoot or get shot), in almost every shot, you can tell the scenes were shot separately by looking at the background. Donald Sutherland's shots are different by the wind hitting the plants in the background; Gene Wilder's shots are different by the position of the sunlight (i.e. one is dark, the other is fully lighted). Obvious if you look closely.
Continuity mistake: If you follow carefully, the twins were not actually mismatched at birth.