PostGIS has a support for geometry with elevation and measurement data. I wonder which GDAL/OGR formats support 3dzm data (if any)? Will I be able to load such data into to database using GDAL/OGR (so far I found only 3d support) and work with it via WFS-T?
Short answer: GDAL does support XYZM geometries but WFS does not.
Support for XYZM in GDAL was implemented by this Request for Comments
It seems to be a bit hard to find from GDAL documentation which drivers currently support XYZM geometries. I believe that at least PostGIS, SQLite/SpatiaLite, GeoPackage, and shapefile does.
With WFS-T you will have heavy problems. WFS is based on GML and while several servers support alternative outputformats I do not know any that would do WFS-T with anything else than GML encoded requests. I would say that if you want to use WFS-T, don't express measures as M-dimension of coordinates. The GML way would be to store measures as measures http://www.datypic.com/sc/niem21/t-gml32_MeasureType.html. That suits well for real measurement data but I do not really know what to do if your measures are related to linear referencing when M-coordinate is logically connected both to the feature (line) and to individual vertices.