How do I harmonize a melody over a passing chord? Especially if the passing chord has notes not found in the key of the song. Should I sing the passing chord tones, and harmonize with other tones found in the passing chord?
A passing chord is simply that - a passing chord. Played, and gone. Usually chromatic to the prevailing key. Certainly not a modulation or a key change. So while we may make a nod towards it, we don't generally embrace the whole chord, as that could give a listener the idea that in fact modulation has in fact occurred.
A passing chord will usually be between two diatonic chords, a sort of link between them. So, it will depend on the actual melody being played/sung at that moment. Sometimes it's best to ignore the notes of that passing chord, often, it's nice to follow along those lines. At least that shows the performer knows what's happening!
It's the same sort of idea with passing notes. They don't have to be reflected in the underlying chord, but they could be, if the performer deems so.