In the young adult science fiction novel EVERLASTING by Holly-Jane Rahlens, a post-disaster future world government suppresses the inconvenience of individualism by abolishing all forms of the first person singular pronoun, replacing them with third-person phrases involving "this _______." For example, a scientist will refer to him/herself as "this scientist." Illeism is actually referenced by name within the plot ó we learn that, earlier in the future society's history, a book written by an unnamed government agency to to teach children this new requirement (and an associated self-repressing philosophy) had been titled "THE ILLEIST'S CODE."
In the Star Wars Expanded Universe, nearly all members of the Gand species refer to themselves in the third person, due to their belief that referring to yourself as "I" assumes everyone knows who you are and is the height of arrogance — unless a Gand has been officially declared notable enough to warrant it.
Young Gand have no names; only after they've done something (say, learning to pilot a ship) do they even get a basic name. Doing something more noteworthy (say, mastering advanced pilot techniques and astronavigation) nets them a personal name, and they have to be truly special to be put through the ceremony that lets them go by "I". Embarrassed Gand use the less specialized names — to make it clearer, Ooryl Qrygg goes by Ooryl normally, Qrygg if he's uncertain or embarassed, and Gand if he's being really humble or has screwed up massively. Being deemed noteworthy enough for "I" just adds another layer; he slips back into third person from time to time.
The author of the part of the X-Wing Series that involves Ooryl, Michael Stackpole, wrote this in the "About the Author" blurb at the end of The Bacta War.
[...] and hates writing these "About the Author" pieces because they force him to refer to himself in the third person. Being neither a Gand nor a presidential candidate, he finds this awkward.
All wolves in The Belgariad speak using "one" instead of "I", or "me", or "you", or... you get the idea. This is supposed to be because wolves have a Hive Mind, but this doesn't bear out in their actual behaviour.
The Reynard Cycle: Glim, the Glyconese ambassador, speaks this way in Defender of the Crown. He refers to himself as "This One."
The Unsullied in A Song of Ice and Fire are an extreme example. They are slaves who have had their real identities literally beaten out of them since childhood. The Good Masters make them pick their names at random each day from a bag, each a combination between a color and a type of vermin. They are forced to refer to themselves as "this one" instead of "I", and given different names each day in order to ensure they will not have any sense of individuality. Like the house-elves mentioned above, the Unsullied live only for their duty.
Shagga, son of Dolf, Strong Belwas and Jaqen H'Ghar are also examples, with H'Ghar being a really interesting case. Not only does he forgo using first person pronouns, but also second person ones, and names in general, choosing to refer to everyone by generic nouns complete with indefinite articles. Instead of saying, "I don't like you," he would say, "A man does not like a girl."
Not really. The book is told in the third person, and told from Cromwell's point of view. Cromwell, however, is almost never referred to by name in the narration; often enough, "he" refers to Cromwell, even though you'd expect it to refer to another character. He does not refer to himself in the third person in dialogue, and there's no reason to believe that he's the narrator.
Formal Chinese etiquette requires those who appear before Judge Dee's bench to use the third person; such as, "This person would like to report a crime." It is generally proper to use the third person when formally addressing a superior.
The titular protagonist of Shane does this towards the end of the book. It is in fact a form of Badass Boast; 'No man should be ashamed of being beat by Shane.'
Fax from Anne McCaffery's Dragonflight does this occasionally as a veiled insult when speaking to F'lar.
The damane in the The Wheel of Time series are required to say their names instead of the pronoun "I", as a means of humiliating/dehumanizing them.
In the semi-dystopia of "Everlasting" by Holly-Jane Rahlens, the first-person singular pronoun was abolished some centuries earlier as part of an ideological war, and is now known only to historians and to the residents of a few vaguely Amish-like enclaves.