No, it is not permitted to use a Kindle or any other electronic e-reader on Shabbat or Yom Tov (Chol Ha'Moed may be a different story).
There are two primary halachic issues with using such a device on Shabbat or Yom Tov:
- The usage of electricity
- The creation of letters
The Usage of Electricity
There is a debate amongst contemporary poskim as to what the prohibition of using electricity is on Shabbat and Yom Tov. Rav Avrohom Yeshaya Karelitz (commonly known as the Chazon Ish) was of the opinion that all applications of electricity would fall into the category of Boneh, the melacha of building and that these applications would be deorita or Torah prohibitions. Rav Chaim Ozer Grodzinski, in his SHu'T Achiezer is of the opinion that electricity falls into the category of ma'avir, the melacha of kindling, and that the Torah prohibition only applies in a situation where heat is generated (ie. a light-bulb) but in other situations it would fall into the category of a Rabbinic prohibition (which may be taken lightly!). Still, other poskim (I believe, Rav Shlomo Zalman Aurbach)are of the opinion that applications of electricity fall into the category of molid or creating something new, which is a Rabbinic prohibition (again, see note above). Rav Moshe Feinstein, in his Igrot Moshe seems to be of the opinion that applications of electricity would not constitute a Torah prohibition, however he cautions not to take this issue lightly as there is a 'chance that it involves the Torah prohibition of maakah b'patish (the final hammer blow)'.
Additional Info: Just found this Wikipedia page which, although I just skimmed it, appears to do a very good job presenting the issue: http://en.wikipedia.org/wiki/Electricity_on_Shabbat_in_Jewish_law
Either way, whether applications of electricity are a Torah prohibition or a Rabbinic one, usage of a Kindle or other e-reader would be prohibited on the grounds that thousands of circuits are created and destroyed constantly as one uses the device.
The Creation of Letters
The prohibition of kotev, the melecha of writing, extends far beyond simply writing letters, the prohibition extends to forming any meaningful images which convey a message (amongst other things). The Mishna Berurah discusses a case in which an individual take a letter made of silver and places it on a background (thus making the letter clear) and rules that this is prohibited as kotev, certainly in the case of a Kindle or other e-reader, where the letters are formed (as opposed to being ready made) and the device is designed to use a contrasting background to make them clear this would fall into the category of kotev.
There will be those who argue that the prohibition of kotev only applies to something which is lasting (there is a debate as to the time-frame required for lasting, but most agree that it is 25 hours), but this argument is flawed. Firstly, this only reduces the prohibition to a rabbinic one (again, see note above). Secondly, I don't think the rule of lasting writing can be applied to this case based on the Rambam in his Mishne Torah where he discussed a case where a person writes on his skin with ink, even though the ink will certainly be erased by the person's sweat, it is nevertheless prohibited based on the fact that at the time of writing it was a permanent act. Based on the fact that the Kindle boasts a one month battery life, I would view each act of 'writing' as lasting for the purposes of hilchot shabbat.
All in all, it is clear, for various reasons, that using any electronic device on Shabbat or Yom Tov is prohibited on either a Torah or rabbinic level and therefore forbidden (except in certain extenuating circumstances which are outside the scope of this post) and certainly adding the element of creating letters to the issue further compounds the prohibition and therefore using a Kindle or other e-reader device would be prohibited.