A dental patient of ours has positional dizziness (for years). He is unmedicated. Is it safe to give him nitrous oxide?

Yes. This sounds like benign paroxysmal positional vertigo. Nitrous oxide will not cause any harm related to that condition.
Absolutely. Positional dizziness is usually due to loose calcium deposits within the inner ear. Nitrous oxide should have no effect at all on this particular problem.
Checkup. Ofcourse, your patient has been thoroughly checked by his physician for a thorough checkup. The gambit could run to uncontrolled blood pressure to a brain tumor. I would try at all possibilities not to use nitrous. If he is not extremely fearfully, just use local anesthesia and gentleness. Take the patients mind off of where he is and talk about his or your favorite subject.