Screen lock settings won't change on Ice Cream Sandwich with MS Exchange Email
Well. First, I deleted my Exchange account. Then I was able to change the screen lock settings and the change would be displayed in the settings page, but when I locked the screen, no PIN or pattern was required to unlock. Then I power cycled the phone. After that I was able to successfully use Screen lock settings. Then I added back my Exchange account, and the phone continued to work fine.

I have no idea what the root cause of my problem was, but it is all working now.