On which trains in Japan is reservation mandatory?

@fkraiem's excellent answer covers most JR trains that are all-reserved. I will add information about all-reserved trains run by non-JR companies.

There are few limited express trains operated by non-JR railroad companies, and a majority of them are all reserved. Those of potential interest to tourists are: ?Sky (Meitetsu), Nagoya-Osaka, Shimakaze (Kintetsu), Hakone (Odakyu) and Rapi:t (Nankai).

• Nagoya Railway (Meitetsu)
  The ?Sky (??????) limited express between Chubu Centrair International Airport and downtown Nagoya (Jingumae, Kanayama, and Meitetsu-Nagoya) is all-reserved and requires a limited express ticket. Some ?Sky trains also travel beyond Meitetsu-Nagoya, to Shin-Unuma or Meitetsu-Gifu

  Note that the regular limited express trains serving Chubu Centrair International Airport are not all-reserved and only require a limited express ticket if you want to sit in a reserved car.

  You can make reservations for Meitetsu trains online.

• Kinki Japan Railway (Kintetsu)
  All Kintetsu limited express trains are all-reserved. These include:

  • the Kintetsu Nagoya-Osaka limited express. There are two types, the "A" limited express Hinotori which stops only at Tsu before arriving at Osaka/Nagoya, and the "B" limited express Urban Liner which makes stops at major stations in Mie and Nara Prefectures.

    This train is more economical than the Shinkansen (4,260 vs 5,940+ JPY), but requires 90 minutes as opposed to 50 minutes on the "Nozomi" Shinkansen;

  • the "sightseeing limited express" Shimakaze, which runs between Kintetsu-Nagoya/Osaka-Namba/Kyoto, and Kashikojima, on Kashikojima Island in Mie Prefecture;

  • the Aoi Symphony, which runs between Osaka Abenobashi and Yoshino;

  • the Ise-Shima Liner, which runs between Kintetsu-Nagoya/Osaka-Namba/Kyoto, and Toba/Kashikojima.

  Some Kintetsu limited express trains also have special cars and premium seats (similar to Green Cars on JR limited express trains), which carry an extra fee (minimum 200 JPY), but much cheaper than the Green Cars. Hinotori, Shimakaze and Aoi Symphony have only special cars.

  You can make reservations for Kintetsu trains online (however, Japanese only).

• Tobu Railway
  All Tobu limited express trains are all-reserved. Those include:

  • the Kegon, Liberty Kegon and Kirifuri running between Asakusa and Tobu-Nikko;
  • the non-regularly scheduled Yunosato running between Asakusa and Shin-Fujiwara;
  • the Kinu and Liberty Kinu running between Asakusa and Kinugawa-onsen;
  • the Liberty Aizu running between Asakusa and Aizu-Tajima;
  • the Shimotsuke running between Asakusa and Tobu-Utsunomiya;
  • the Ry?m? running between Asakusa and Akagi;
  • the Skytree Liner running between Asakusa and Kasugabe;
  • the Urban Park Liner mainly running between Asakusa and Omiya.

  There are Tobu trains to the popular tourist destinations Nikko and Kinugawa, but usually the "Nikko" and "Kinugawa" trains operated by JR East (but onto Tobu lines) are more convenient because they depart from Shinjuku.

• Seibu Railway
  All Seibu limited express trains are all-reserved:

  • the Chichibu running between Ikebukuro and Seibu-Chichibu;
  • the Musashi running between Ikebukuro and Hann?;
  • the Koedo running between Seibu-Shinjuku and Honkawagoe.

• Keisei Electric Railway: the only Keisei limited express train, the Skyliner, is mentioned above and is all-reserved.

• Odakyu Electric Railway
  All Odakyu limited express trains (branded "Romance Car") are all-reserved:

  • the Hakone and Metro Hakone, running between Shinjuku/Kita-Senju and Hakone. If you plan to go to Hakone, chances are that you are going to take this train;
  • the Enoshima and Metro Enoshima, running between Shinjuku/Kita-Senju and Katase-Enojima. There are a few ways to get to Enoshima but this is the only direct train;
  • the Morning Way and Home Way, running between Shinjuku/Otemachi/Kita-Senju and Hakone-Yumoto. These are commuter expresses probably of no interest to tourists.

  You can make reservations for Odakyu trains online.

• Nankai Electric Railway

  • the Rapi:t running between Namba and Kansai International Airport (however, note that its JR West counterpart, the Haruka, has non-reserved seats);
  • the K?ya running between Namba and Gokurokubashi;
  • the Rinkan running between Namba and Hashimoto;
  • the Senboku Liner running between Namba and Izumi-Chuo.

• Sagano Scenic Railway
  Sagano Scienic Railway runs trains on the Sagano Sightseeing Line using open-air trains. All trains are all-reserved. You may make reservations on the JR West website, or at major JR West stations.