I recall that some of the supporting characters' dialogue got cut in the Director's Cut too.
Not much. Some of the supporting character's dialogue had been
left on the cutting room floor for the theatrical version, but finally restored for the ABC-TV "Special Longer Version" (and subsequent release on VHS), such as the bridge Rhaandarite's defence of Captain Decker (which had been featured in the Marvel Comics adaptation and in the novelization) and his admonishment by Uhura. That scene was retained for the DVD DE.
As a fan of Billy Van Zandt, it's the scene I miss the most on the recent BluRay version, which goes back to the original theatrical version.