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: μSky (Meitetsu), Nagoya-Osaka, Shimakaze (Kintetsu), Hakone (Odakyu) and Rapi:t (Nankai).

• Nagoya Railway (Meitetsu)
  The μSky (ミュースカイ) 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 μSky 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 Ryōmō 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 Hannō;
  • 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 Kōya 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.

First a reminder: whether one posseses a rail pass such as the Japan Rail Pass has no effect on seat reservation requirements. If reservation is required, it remains so if you have a rail pass. Of course, a rail pass usually allows you to make reservations free of charge, but it is still necessary to visit the ticket office to make your reservation. (Note that some minor regional passes do not allow you to make reservations; in that case all-reserved trains require payment of at least the reservation fee.)

Firstly, there are four all-reserved Shinkansen services. Note that all services on the Tokaido-Sanyo-Kyushu Shinkansen between Tokyo and Kagoshima-Chuo, serving Nagoya, Kyoto, Osaka, Hiroshima, Fukuoka, Kumamoto, etc. have non-reserved seats.

• The Hayate and Hayabusa services on the Tohoku-Hokkaido Shinkansen between Tokyo and Shin-Hakodate-Hokuto. However, if your trip is fully on the section between Morioka and Shin-Hakodate-Hokuto, see the note at the bottom.
• The Komachi services on the Akita Shinkansen between Tokyo and Akita. However, if your trip is fully on the section between Morioka and Akita, see the note at the bottom.
• The Kagayaki services on the Hokuriku Shinkansen between Tokyo and Kanazawa.

If you try to make a reservation on one of those trains and the train is fully booked, you will have the option to obtain a standing ticket, which as its name implies lets you ride the train by standing in the intervals between carriages. A standing ticket is slightly cheaper than a reserved seat ticket, as it does not include the seat reservation fee. It is not possible to obtain a standing ticket in any other circumstance, such as if the train is not fully booked (in order to save money) or on any train other than the four above.

The two “premium” express services serving Narita airport are also all-reserved:

• The JR Narita Express, serving many stations in the greater Tokyo area. However, see the note at the bottom.
• The Keisei Skyliner, serving Nippori and Ueno.

The two surviving regularly scheduled overnight trains:

• The Sunrise Seto, between Tokyo and Takamatsu.
• The Sunrise Izumo, between Tokyo and Izumoshi.

Finally, almost all “touristy” trains are all-reserved, such as:

• The Resort Shirakami, along the Sea of Japan coast of Aomori and Akita prefectures.
• The POKÉMON with YOU, between Ichinoseki and Kesennuma.
• The Yufuin no Mori, between Fukuoka and Beppu.
• The SL Hitoyoshi, between Kumamoto and Hitoyoshi.
• and many more…

The easiest way to find out whether a given train has unreserved seats (or, in general, which seating options are available on it) is to look it up on Hyperdia, and see what options are available in the drop-down list of the “Seat Fee” column. If the column is empty, it means the train is an ordinary commuter train, where reservation is not possible in the first place.

Note about the Hayabusa, Hayate and Komachi services beyond Morioka. To compensate for the fact that no services with non-reserved seats exist beyond Morioka, it is possible to board those services without a reservation, by purchasing a “special limited express ticket” (特定特急券) instead of a regular reserved-seat limited express ticket. A special limited express ticket works in essentially the same way as a non-reserved-seat one would: it is cheaper than a reserved-seat one since it does not include a seat reservation fee, and it allows the holder to use any vacant seat, with the caveat that if another passenger has a reserved-seat ticket for that seat, the latter has priority. Rail pass holders, meanwhile, can simply board with just their pass.

Note about the Narita Express. Similarly, the Narita Express can be boarded without a reservation, by purchasing a “non-designated seat ticket” (座席未指定券) instead of a regular reserved-seat limited express ticket (or just boarding with a rail pass). As above, a non-designated seat ticket allows the holder to use any vacant seat, with the caveat that reserved-seat ticket holders have priority. However, a non-designated seat ticket costs the same as a reserved-seat one. A non-designated seat ticket can also be converted to a reserved-seat one at no extra cost.