Updated findings on the patch 1.6 dialogue bug from personal testing:
Dialogue in a UI menu (actually talking to someone)
Only the audio is failing to play. If your subtitles are on, they will display and run for the length of time they're supposed to. Any scripts attached will execute once the dialogue is finished displaying. I've verified this with my Paarthurnax mod.
Dialogue outside the UI menu (including scenes)
Audio, dialogue, AND any scripts completely fail. This will cause things like the courier to show up attempting to deliver a letter and being unable to (if the mod edits courier dialogue that is) and with things like the Jagged Crown scene and Armen's sword scene to fail to run properly. So basically any "hellos" from NPCs and anything used in a scene.