Have you tried deactivating all your plugins and switching to the twentytwelve theme to start ruling out causes?
Are there any Javascript errors? Try using Chrome’s debugger or similar. (Hit F12 in Chrome on Windows and look for a red error sign in the bottom right hand side)
Depending on what theme you’re using, you may not have a responsive navigation built in.
If you don’t then consider using a standalone one that you can add, such as Bootstrap’s.
If you have edited the wp_nav_menu call then make sure you have not removed any class names, as it’s these that the JaaScript hooks on to to show/hide the menu and its button.
If the button is working and there are no JavaScript errors, then double check you haven’t removed any CSS classes that would control the show/hide effect.
If you’re stuck, just provide a URL and I’ll let you know the issue you’re having.
