Several people have already posted about the other two incidents involving "thee" and "thou": Denethor's final conversation with Gandalf in "The Pyre of Denethor", and the Mouth of Sauron's words to Aragorn and Gandalf in "The Black Gate Opens". Since the usage of the familiar personal pronoun is similar in these two passages (and completely different from Éowyn's use), we can discuss them without pause.
One of the uses of the familiar personal pronoun is talking down to other persons. That takes place when non-family adults speak with children, for example, but also in less friendly situations like when some people speak down at foreigners as if they were not adults. And of course it also can take place when one person insults another and uses the familiar form - in German, for example, saying "Du Idiot!" to a stranger.
I find it very noteworthy that Denethor, who was always polite previously, turns insolent in regard to Gandalf at the end of his life. He rejects any kind of authority the wizard might have, and though he may not know precisely which nature Mithrandir has as a Maia, he must realize that the wizard is higher than himself. But he lashes out at him, "Since when has the Lord of Gondor been answerable to thee?" He accuses Gandalf of attempting to make him a tool for his own purposes. The Steward no longer wants to be under other authority and expresses that violently, using "thee" and "thou" to emphasize his point.
Christopher Tolkien speaks about this usage briefly in HoME 12,
The Peoples of Middle-earth and quotes his father:
Quote:
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Where thou, thee, thy appears it is used mainly to mark a use of the familiar form where that was not usual. For instance its use by Denethor in his last madness to Gandalf, and by the Messenger of Sauron, was in both cases intended to be contemptuous. But elsewhere it is ocasionally used to indicate a deliberate change to a form of affection or endearment.
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The last sentence must refer to the Éowyn passage.