I remember reading an article (linked from HN I believe) that was a response to the "Waiting for Superman" movie which cited numbers showing that private schools didn't perform any better than public schools. That would suggest that either the ability to fire teachers subjectively doesn't have an impact on student performance, or that private schools don't fire enough teachers or the metrics the private schools use to fire teachers are not in line with what would improve student achievement.
(That's a Jerry Weinberg use of the j-word, "just". It would be a lot of trouble to do.)