Why are geodesic airframes no longer produced? Do they cost more than conventional structures? Are they not as strong as conventional structures? I assume it is insufficient cost benefit ratio, but how does that work out?
Geodesic (sometimes also spelled geodetic) designs were used in aviation as early as 1909, in the Schütte-Lanz SL1 airship. It had a wooden structure with fabric covering, and Professor Johann Schütte, the scientific head of this design, used the most efficient method conceivable. However, planning the shape of all structural members was an enormous amount of work, and stretching an existing ship by inserting a new section was almost impossible. For those reasons, the late Schütte-Lanz airships used aluminum frames with conventional design, just like Zeppelin did from the beginning. Diagonal wire bracing was used to take up shear loads.
Geodesic frames are good in transferring both bending and shear loads. This is helpful when the aerodynamic covering is not stiff enough to contribute any shear stiffness, as in the case of fabric covering. However, when the skin is made from the same material as the frame, it can carry shear loads and a geodesic frame would not improve matters. Now it is better to carry longitudinal loads in longerons and shear in the skin.
But geodesic constructions are not dead. CAD makes the design effort manageable, and in some cases they give a weight advantage. Look at the casing of the EJ200 engine: It uses a geodesic reinforcement to make the casing light and strong and to prevent harmful vibrations. In the end, a jet engine casing is a big pressure vessel and needs to be strong, stiff and light.
The switch from internal frames to structural skin airframes -- called monocoque construction -- is fairly well covered in Wikipedia:
Early aircraft were constructed using internal frames, typically of wood or steel tubing, which were then covered (or skinned) with fabric3 such as irish linen or cotton. The skin added nothing to the structural strength of the airframe and was essentially dead weight beyond providing a smooth sealed surface. By thinking of the airframe as a whole, and not just the sum of its parts, it made sense to adopt a monocoque structure and it did not take long for various companies to adopt practices from the boat industry such as laminating thin strips of wood
It started when the usual way of building was a steel-tube frame, with wooden formers attached to carry the skin. Early geodetic planes were still fabric covered, but the entire airframe structure was now in the skin, leaving the interior open for payload like people or ordnance. It was much lighter and stronger, though patented and many builders simply don't want to license-build as much of their plane from somebody elses' ideas.
Once metal skins were being used, it wasn't the only monocoque option, though it was still superior to the usual orthogonal structures. Still more expensive though.
"Geodesic" should refer to the field of mathematics itself. Geodetic applications of it work for space-frame structures or mapping a globe, etc.
Geodesic airframes are useful for purpose-built crop dusters, built from 4130 stainless steel tubes. The chemicals on board could easily react with aluminium, which is still used for skin panels but not as a structural member: the skin plates are only for covering, not for taking any of the prime loads. The construction method is one of the reasons why they look unlike any other plane, with the angular cockpit shape and all.
Other than that, monocoque construction has completely taken over from geodesic frame construction - when the skin plates are the main load bearers the airframe is lighter.
As other answers already note, the major attraction of geodetic framing was to provide a lighter and stronger structure than internal framing, with the not inconsiderable bonus of leaving much more internal space for fuel and payload. The geodetic frame could directly carry a fabric skin in a streamlined shape, instead of requiring the addition of (heavy) falsework merely to carry the skin.
Some modern aircraft are still built in a frame-and-fabric manner, such as the Aviat Husky, though most use a more generic "space frame" construction rather than geodetic framing. These are almost universally General Aviation aircraft designed principally for VFR flight.
A general disadvantage of fabric-skinned aircraft is that they do not easily support cabin pressurisation; it would be necessary to install a second, internal pressure hull to retain a sufficient air seal. While this would not be especially difficult to achieve, it does reduce the main advantage of the geodetic frame, in the context of aircraft designed to regularly fly above about FL120 (where most people start to experience debilitating hypoxia) without carrying oxygen for every occupant for the entire cruise phase of the flight.
Instead, metal-skinned and composite aircraft are typically designed so that the skin forms the pressure cabin as well as the aerodynamic surface, and also carries some of the structural loads so that less internal framing is required.
One area where geodesic airframes have seen a recent resurgence, is 3-d-printed model airplanes.
I've seen an example where the fuselage and wings and tail surfaces were all of open geodesic construction, similar to the well-known Vickers "Wellington". The model was covered by a heat-shrunk plastic film such as Monokote.
Here's a website featuring many examples of such models-- https://www.3dprintedrcplanes.com/
This link describes the process of designing and 3-d-printing double-walled structures for model airplanes, with an internal geodesic structure between the inner and outer walls-- https://www.rc3dprint.com/post/3d-printing-geodesic-internal-structure
They ARE still produced. See for example the EU research project from 2013 here: http://www.transport-research.info/project/advanced-lattice-structures-composite-airframes