That being said, he does owe his students the chance to at least look at the culmination of their semester's work. I know at my university, professors are obligated to have exams available for at least a year past the date they were taken! Maybe if allowing students to look at their exams would truly take an additional 75 hours of his time as claimed, he'd have a stronger case, but that's simply not true. If the student was requesting him to explain why a problem was wrong, or engage with this exam in anyway, I'd be completely in agreement. Hell, he could just put them all in a box outside his door, or maybe leave them with the departmental secretary (this happens all the time), or come up with some sort of solution.
His broader point still stands, but this is absolutely part of the responsibilities of a teacher, and I can't get behind this guy making a point at the expense of his students.