The use of the formal "thee," "thou," and "thy" in Hemingway's novel may serve three purposes. First, Hemingway meant his dialogue to be a direct translation from Spanish, since the novel is set in Spain during the Spanish Civil War. In Spanish, the pronoun you has two forms, the formal usted and the familiar tú. Throughout the novel, Hemingway uses odd translations and, of course, censors the profanity which is common in the dialogue of the Spanish characters (he simply replaces the profanity with words such as "obscenity" and "muck").
Second, Hemingway uses these formalities to suggest the camaraderie of the characters who are fighting for a cause they are willing to die for. Rather than referring to each other as señor, señora or señorita, the characters refer to each other on equal terms as thee and thou. It is similar to the communist term comrade which is also a term of equality. The rebels during the civil war were attempting to bring freedom and equality to Spain, in contrast to their fascist opponents.
Third, the terms may be an allusion to religious symbolism and the Bible. Patrick Cheney, in his article "Hemingway and Christian Epic: The Bible in For Whom the Bell Tolls" argues that the novel is replete with Christian imagery, including the constant use of the biblical "thee" and "thou" (of course, these words were first used in the English translation, the King James Bible). An example of this imagery and symbolism is the description of El Sordo's death and the implication that he is a martyr for the cause of the Spanish Republic. He dies on a hill and in a direct reference to Jesus Christ (who was crucified on Calvary Hill), Robert Jordan says,
If he had known how many men in history have had to use a hill to die on it would not have cheered him any for, in the moment he was passing through, men are not impressed by what has happened to other men in similar circumstance...
Thus, all of the thees and thous give Hemingway's work a biblical effect and may even suggest that Robert Jordan and the Republicans he is fighting with are on the side favored by God. Hemingway, however, would certainly recoil from this link, as he did when critics used biblical references in their analysis of The Old Man and the Sea.