This question is a little tricky because it conflates two concepts. The first is POSTing and the second is Ajax. POSTing, as compared to GETting, is a different HTTP method with different implementation and semantics. Ajax, or XmlHttpRequest, has its counterpart in normal requesting/navigation. You can use POST or GET for both XmlHttpRequest and normal navigation and form submission (well, POST is always a "form submission"). The difference between XmlHttpRequest and normal requests is that a normal request replaces the page with a new page.
You could write just about any website using only Ajax to get data and change the DOM; that's mainly how Gmail works. There are no "form submissions" in the traditional sense. But there are still GETs and POSTs because the server and browsers interpret the results differently. GET is supposed to be idempotent; POST is meant for operations that change the state on the server. For example, an ecom transaction should be a POST. This doesn't change when using Ajax because you want proxy servers to also understand that this is a POST and that they shouldn't try to cache the response.
There are also advantages and disadvantages to using GET vs POST. You can't bookmark the results of a POST because the parameters are hidden. You can't GET something with parameter values of unlimited length because IE only supports about 2000 chars.
Also there are disadvantages to using Ajax vs normal submissions; you can't bookmark the resulting page; the back button won't work as expected. You can't upload a file using Ajax. But with Ajax you could minimize the data transfered and also provide convenient updates to a page (such as monitoring the status of a long process) without annoying flickering or refreshing.
In summary the two request types, Ajax and traditional form submission, can both be used with GETs and POSTs, and there are pros and cons for each. Neither type can do everything the other can and so you can expect to see a mix for the forseeable future.