When it hangs, try to see what is being logged in your browser's console. This is usually accessible via the Developer Tools (F12).
In the console tab, see if there are any errors.
Does it happen when posts or thread titles contain specific words? We sometimes see things like mod_security blocking replies or thread creation if the thread title contains certain words it doesn't like. Similarly, things like adblock in the user's browser can prevent posts being submitted if they contain references to advertisements.
How frequently does it happen? Is there anything in any Apache log (either a general one or VHost specific)? Are you using mod_php?
Ideally, we would need to see what response is actually happening. More generically, assuming Chrome, you'd need to have the dev tools open to the network tab and browse with that open until it happens. Then we'd need to find the Ajax request in question to see what sort of thing is happening.