The devices Steiner uses here have their source in 17th century practice. At that time, the element creating the dissonance was a pedal point (stationary bass), over which you could play all kinds of chords, consonant or dissonant, fitting the scale or chromatic. Here's a simple example from J. S. Bach's Little Prelude in C Major, BWV 924:
The end of this pedal point passage shows the other element -- the resolution. Note that the 4 (G in the bass with C above) resolves at the last moment to 3 (part of the dominant chord) before the final tonic sounds.
The resolution element isn't always present -- the dissonance can be left hanging, so to speak. In the reunion scene, when Rick and Ilsa first see each other in Casablanca, there is an independent dissonance of this kind when Ilsa says "That's when the Germans marched in." The E# diminished seventh chord could easily settle into an F# major or minor triad, but Steiner just lets it fade off -- in 1943, after all, the Germans hadn't marched back out yet.
The famous chord at the beginning of the reunion scene -- what James Buhler and I call the "gaze sonority" -- at (a) -- is even more complicated, though, like "the Germans marched in," it too fades rather than resolves. This chord is a full D minor triad with an Em7(flat5) tucked in the middle -- see at (b). Joel Love has pointed out that this is a possible voicing of a C13 chord, except that the root of the chord is missing! -- at (c). The better voicing is the second one, though, with the more typical #11. We hear that version (again without bass) when Ilsa returns to Rick's Café after hours.