Casablanca has a narrator, but he's only around for about 2.2 seconds at the very beginning to set the stage for us. He doesn't even last as long as Ugarte.
Since there's some historical background that a viewer should be aware of, the movie makes use of this narrator to fill us in on the important details, so that it doesn't have to be forced into the characters' lines as blatant exposition.
And while it doesn't go crazy with the flashbacks, one of the most memorable scenes (okay, montage of scenes) in the movie is the flashback to Rick and Ilsa's time in Paris, when they'd first met and were madly in love. This is way more effective than simply hearing them talk about it, because we get to grab a glimpse of Rick before he had the weight of the world on his shoulders, and Ilsa before she was weighed down by all that guilt. Come to think of it, these two are dealing with a ton of extra weight. Might be time to call Jenny Craig.