I understand that the maximum length for an AJAX call using GET is 2083 (at least with IE6.)
Is there any such limitation when doing an AJAX call using POST? I've never used a POST with an AJAX call so I'm not even sure what other differences there might be.
That depends on your server.
IIS/ASP.Net (by default) has a limit on the size of a post body, (about 2 MB). But this can be changed using the configuration files.
For normal AJAX usage, you will not run into trouble. If you start uploading files, then you will reach the limits very easy.
No, this is no arbitrary byte limit to posts. However sending huge amounts of data up or down from a web application could cause problems with users, especially if they have a capped bandwidth plan.