You need to look at the GetCapabilities response of the GeoServer services to see if those services support XML/POST requests for GetMap requests. For example in one GeoServer service I just looked at:
<OnlineResource xlink:type="simple" xlink:href="http://xxx/geoserver/CPRM_EN_Bedrock_and_Age/ows?SERVICE=WMS&"/>
Only <Get> support is listed.
I looked at a couple of GeoServer services and didn't see Post listed, so its possible that XML/POST support is not available in GeoServer as a default operation.
As your question is specifically about GeoServer I should probably stop there, but I can tell you that other FOSS4G WMS server software does support XML/POST operations for GetMap requests, for example MapServer