During his concert, Henry's hair changes back and forth from neat to messed up.
When Henry takes a taxi from a restaurant, the establishing shots clearly show that it is less than half a block to a T-shaped intersection. However, a moment later, the interior shot of Henry in a panic as the taxi drives off shows a noticeably longer stretch of uninterrupted street outside the windows.
When Peter Sellers is rifling through the phone book looking for an airline he comes down to flipping single pages then suddenly flips a bunch of pages and immediately finds the number.
The film story supposedly starts in late autumn/early winter with leaves coloring and blowing about and "the holidays" coming up, but the calendar in Gil's kitchen is for January.
When Mrs. Gilbert pours Mr. Boyd a drink at her home, the "scotch" foams slightly. Real booze doesn't do that; its ubiquitous stand-in, cold tea, does.
As Orient drives off in a taxi from the "spaghetti joint", he sees Val and Gil in the rear view mirror. But the image of the girls and the street are not reversed as they would be in an actual mirror.
The balcony audience directly behind the conductor are clearly cardboard cut-outs.