This is now possible using just selections and a rule-based symbology.

This requires the Expression Plus plugin.

Create a rule based symbology with the filter of:

QGIS 2.12+:

isselected( @layer_name )

Or QGIS 2.8:

isselected( 'ACTUAL_NAME_OF_LAYER' )

Now only the selected features that meet this rule will be shown.

You can also have an ELSE rule for all unselected features.

This is now possible using just selections and a rule-based symbology.

This requires the Expression Plus plugin (for QGIS 2.x. 3.0 has the functionality built in).

Create a rule based symbology with the filter of:

QGIS 3.0+ (from thequerist in the comments):

is_selected()

QGIS 2.12-2.18:

isselected( @layer_name )

Or QGIS 2.8:

isselected( 'ACTUAL_NAME_OF_LAYER' )

Now only the selected features that meet this rule will be shown.

You can also have an ELSE rule for all unselected features.

This is now possible using just selections and a rule-based symbology.

This requires the Expression Plus plugin.

Create a rule based symbology with the filter of:

QGIS 2.12+:

isselected( @layer_name )

Or QGIS 2.8:

isselected( 'ACTUAL_NAME_OF_LAYER' )

Now only the selected features that meet this rule will be shown.

This is now possible using just selections and a rule-based symbology.

This requires the Expression Plus plugin.

Create a rule based symbology with the filter of:

QGIS 2.12+:

isselected( @layer_name )

Or QGIS 2.8:

isselected( 'ACTUAL_NAME_OF_LAYER' )

Now only the selected features that meet this rule will be shown.

You can also have an ELSE rule for all unselected features.