Not all vendor implementations support all parameters, one of the joys of working with OGC services. Also note that GeoServer implements WFS 1.0.0, 1.1.0 and 2.0.0 (not 1.2.0 in your sample above)
You can do what you want with a (relatively verbose) POST request using an OGC filter, this might work with a GET but I'll leave that as an exercise for the reader...
<wfs:GetFeature
xmlns:ogc="http://www.opengis.net/ogc"
xmlns:gml="http://www.opengis.net/gml"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xmlns:xlink="http://www.w3.org/1999/xlink"
xmlns:ows="http://www.opengis.net/ows"
xmlns:wfs="http://www.opengis.net/wfs"
xsi:schemaLocation="http://www.opengis.net/wfs http://schemas.opengis.net/wfs/1.1.0/wfs.xsd"
service="WFS" version="1.1.0" outputFormat="JSON">
<wfs:Query typeName="it.geosolutions:tsige" srsName="EPSG:4326">
<ogc:Filter>
<ogc:And>
<ogc:PropertyIsGreaterThanOrEqualTo>
<ogc:PropertyName>your-time-variabe-here</ogc:PropertyName>
<ogc:Function name="dateParse">
<ogc:Literal>yyyy-MM-dd</ogc:Literal>
<ogc:Literal>2011-07-25</ogc:Literal>
</ogc:Function>
</ogc:PropertyIsGreaterThanOrEqualTo>
<ogc:PropertyIsLessThan>
<ogc:PropertyName>your-time-variable-here</ogc:PropertyName>
<ogc:Function name="dateParse">
<ogc:Literal>yyyy-MM-dd</ogc:Literal>
<ogc:Literal>2011-07-26</ogc:Literal>
</ogc:Function>
</ogc:PropertyIsLessThan>
</ogc:And>
</ogc:Filter>
</wfs:Query>
</wfs:GetFeature>
You could test this using curl with the request saved as wfs.xml
curl -d @wfs.xml -H "Content-Type: application/xml" "http://localhost:8082/geoserver/it.geosolutions/ows"