Definetly something is wrong in your server configuration since the ajax requests aren’t being executed.
Even the simplest ajax request WordPress does by default aren’t working.
Check with your hosting support why this is happening because it’s not normal.
This is an example of ajax call that WordPress does by default. You can share with them this image.
https://www.dropbox.com/s/1a3m5x6dqnzqi7q/Screenshot%202016-05-24%2011.02.33.png?dl=0
Also check your server error logs they usually provide additional information about the errors.
Let us know the result.
-Rui