What are the abandoned technologies in aviation industry
The biggest one that comes to mind for me is Cathode Ray Tube technology for cockpit displays.  Rockwell Collins (called something else now) has ceased production of CRT displays and users of Proline CRT Primary Flight Displays, Secondary Flight Displays and EICAS systems are in a scramble to replace the CRTs when they wear out.  There are a number of companies offering LCD units to upgrade CRT units under STC and this is happening across the airline industry.

Another one could be navaids like Non Directional Beacons that are starting to go the way of the A/N Range and the related receivers, and eventually other analog navigation equipment.

Mechanically, not that much.  Light aircraft are still built today with magnetos that use points and condensers like tractors from the 40s.  They are simple and work.  The business is very conservative, a vital self-preservation characteristic in such a life-or-death environment, and things that are simple, work, and are easy to fix and easy to diagnose, and let you know when they are degrading, are hard to replace.