After some digging through links provided by @Heopps, here are some specific examples.
Note that agencies often do not draw attention to their obsolete technologies until they have already replaced them (or at least have plans to replace them), so some of the items below may have already be discontinued. Also, there are likely to be more obsolete tecnologies that are not publicized.
Geolocation by ground stations (instead of GPS/GLONASS)
Although sat nav is a common sight in cars on Earth, Soyuz until now relied on six ground stations for precise measurements of its orbital path. With Soyuz-MS, engineers will do away with a bulky system in favor of a new "Apparatus for Satellite Navigation" or ASN-K, which can talk to GPS satellites and their Russian counterparts known as GLONASS.
Analog television (instead of digital)
A new digital television system, TVS, which replaced an older Klest (crossbill) analog TV, allows transmission between the transport ship and the space station via onboard radio channels.
Analog signal processing (instead of digital)
The Kurs-NA has been in development since 2003 under leadership of Sergei Medvedev at Moscow-based AO NIITP, the division of AO RKS corporation. In the course of the project, team aimed to "almost entirely" drop analog signal processing, switching to fully computerized operation performed by a powerful three-processor machine capable of self-diagnostics.
Analog avionics (instead of digital)
A number of avionics went out of production in the Motion and Orientation Control Unit, BUPO, (from Russian Blok Upravleniya Peremesheniem i Orientatsiei). The latter was replaced with a new digital control unit dubbed Blok Upravleniya Rezervnym Konturom, BURK, which can be translated as the Backup Loop Control Unit.
Halogen lightbulbs (instead of LED)
By 2010, the Lisma-Postarm company in Saransk, Russia, could no longer guarantee the supply of KGZ 12-100 halogen light bulbs for the SMI-4 lights, which were used to illuminate the target during docking. Instead, the MS series received a new LED-based lighting system, SFOK, which was first tested on the Progress-MS-01 spacecraft.
Two-piece fuel injectors (instead of one-piece)
The older RD-107 (11D512) and RD-108 (11D511) engines used 260 two-component centrifugal injectors, while the new engines received more than 1,000 one-component injectors.
(In fact, the rocket model FG is named for the abbreviation for the Russian words for "injector head".)