Are there any trains that require extra fees with a Japan Rail Pass ?

With a Japan Rail Pass, almost all of your journeys on JR trains are covered in the pass price. However, there are a few exceptions. Travelling on a JR train that uses a non-JR railway line for part of the journey requires you to pay an extra fee for the section operated by the non-JR company. Payment can be made on the train, or at the station. Some journeys where this happens are below.

TO GET TO KAWAGUCHIKO:
If you ride the FUJI EXCURSION train between Shinjuku and Kawaguchiko, the train partly uses the private Fujikyuko Line, so you will need to pay an additional charge of around 1200 yen.

TO GET TO AMANOHASHIDATE:
To reach Amanohashidate, you will have to ride on the Kyoto Tango Railway. From Kyoto to Amanohashidate/Toyooka the additional fare is about 3000 yen, from Shin-Osaka to Amanohashidate it's 2000 yen.

TO GET TO TOTTORI:
To go to Tottori from Kyoto or Okayama, the Super Hakuto and Super Inaba trains use a private section of train line between Kamigori and Chizu. An additional fare of around 2000 yen is required.
The same applies to the Super View Odoriko and the Odoriko, a direct Express train from Tokyo to Izukyu Shimoda, from Nagoya to Shingu with the Wide Express Nanki, or from Nagoya to Toba with the Limited Express Rapid Mie, from Shinjuku to Gotemba, with the Asagiri train and in some other very few lines.