Steam locomotive is called SL in Japan and JR does not run any SL trains for regular timetable.
On the other hand, there are several lines in Japan, including JR where SL trains are running as special trains.
The most famous one is JR Yamaguchi line in west Honshu, between Shin-Yamaguchi in Yamaguchi prefecture and Tsuwano station in Shimane prefecture. Shin-Yamaguchi station is on the Sanyo Shinkansen line and Tsuwano city is a historical town called Small Kyoto.
Private railways also provide SL trains for non-regular basis. The famous among them are Chichibu railway and Oigawa railway. Chichubu railway's SL are runnning between Kumagaya station and Mitsumineguchi station. Kumagaya station can be reached by JR Takasaki line or Joetsu or Hokuriku Shinkansen lines. The SL also stops at Nagatoro station which is close to the Nagatoro gorge where you can enjoy the boat river cruise. Oigawa railway's SL are running between Shin-Kanaya and Senzu. Please be careful that Shin-Kanaya is not connected with JR line. Oigawa railway has its station connecting with JR Kanaya station but the SL does not depart from Kanaya station. Anyway, you can take a normal train to Shin-Kanaya and then change to SL train.
SL trains once dominated the transportation of Japan but almost disappeared in early 80s. Although it is extremely popular to young kids but I assume that a lot of the Japanese do not know how the steam locomotive moves. Then it is ironic that people hate the smokes and the sounds in 70s but getting excited when they see the SL train moving in 21st century.