GDAL supports two SQL dialects: the default OGR dialect and an alternative SQLite (Spatialite) dialect. Both dialect support selecting geometries by geometry type. Ogrinfo tool is recommended for experiments because by using it the possible troubles with writing the result out can be avoided. If output from ogrinfo is good then the reading side is OK too.
Making a selection by geometry type with the OGR SQL dialect:
ogrinfo -sql "select * from entities where Layer='New' and OGR_GEOMETRY='POINT'" dxftest.dxf
Making a selection by geometry type with the SQLite SQL dialect:
ogrinfo -dialect sqlite -sql "SELECT * FROM entities WHERE layer='New' and GeometryType(geometry)='POINT'" dxftest.dxf