Please try this. Open your website in Google Chrome, open the DevTools window (using F12), and switch to Console tab. Then try opening the Customizer and try saving changes. Do you see any errors?
Also, if you switch to the Network tab on DevTools and filter by XHR, do you see any ajax request that have an error? If so, can you inspect that and update your question?
If everything is fine on the DevTools, you should check the PHP version of the server. If it is lower than 5.4, there is high probability that this is the cause. Nothing else comes to my mind.