I have seen online discussions of natural languages that avoid the use of "I" and of constructed languages (outside a literary context) that don't even have pronouns (Elkaril, apparently). See for example the Straight Dope Message Board and Language Log. Some of them mention Samuel R. Delaney's novel Babel-17, which was published in 1966, as an example of a novel that containns a language without the pronoun "I". From Wikipedia's description:
The language portrayed at the center of Babel-17 contains interesting linguistic features including the absence of a pronoun or any other construction for "I". The heroine finds her perceptions (and even her physical abilities) altered once she has learned Babel-17. In this Delany's novel influenced a generation of writers: Native Tongue by Suzette Haden Elgin, The Dispossessed by Ursula K. Le Guin, Embassytown by China Miéville, "In Luna Bore Coda" by Joshua Nilles, and, more evidently, the short story "Story of Your Life" by Ted Chiang. It also resembles a few preceding science fiction novels which deal with how languages shape the political and cultural stratum of societies, such as The Languages of Pao by Jack Vance or Anthem by Ayn Rand, and language as a weapon was adapted as a plot device in Neal Stephenson's Snow Crash.
Some of the claims about influence on other authors require citation that provide evidence. The last sentence mentions two other works to look at, but The Languages of Pao, first published in 1957 or 1958, does not appear to contain a language in which using the pronoun "I" is prohibited.
In Ayn Rand's novella Anthem, published in 1938, singular pronouns are totally absent. The article The Style and Literary Form of Anthem explains,
One of the first things we discover reading the novel is that the narrator never uses the word “I” when he is referring to himself. He always refers to himself as “we” and he uses the words “us” and “our” when he is talking about himself. (...)
As we read on in the story, we discover that this is a feature of the narrator’s entire society. Nobody ever uses any singular pronouns. “I,” “him,” “her”—none of these words are ever used. All they ever use are plural pronouns—“we,” “our,” “they,” “them.”
The world of Anthem is a world in which the very word “I” and all other singular pronouns seem to have disappeared completely from the language. And because the people in this society can’t speak of themselves as distinct, single individuals—they don’t have the word to talk about themselves as distinct, single individuals—they can’t think of themselves as single, distinct individuals.