Messages.app asks for iChat every time I open it
(I haven't upgraded to Mountain Lion myself, but my understanding is that Messages has subsumed iChat.) That looks like the AppleScript find-the-application-named-in-the-script dialog. Do you perhaps have something that somehow runs a script when you launch Messages? If so, find that script and edit it to refer to application "Messages" instead of application "iChat" if that works, or remove it entirely.