If changes are not saving or you cant get confirmation when email is sent then there is a problem with the JSON object the scripts return. Some of the things that can go wrong are:
JSON was added in PHP 5.2. Even if PHP is 5.2 or newer(WP requirements) make sure it wasn't compiled with --disable-- json.
Warnings and notices should be disabled in the production server. Even though, all PHP warnings/errors/notices are turned on in our development machines we cannot produce any errors/warnings/notices. If PHP notices are enabled, that might cause a broken JSON object being returned.
There may be other plugins/themes that are causing Errors(Someone else had similar problems and their site was filled with JS errors).
P.S. Browser plugins (popup blocker, flash blocker, ad blocker, ...) inject extra scripts in the pages. Make sure there are no conflict there either.