The is pronounced "thee" when it precedes a word that begins with a vowel (the apple, the overtone series, etc.) or (sometimes) an aspirated consonant (the historic occasion of his birth) or when the speaker wishes to differentiate a noun by calling it out for special dramatic emphasis.
For example:
"He was the heavyweight boxing champion."
"He was the heavyweight boxing champion."
The first is a simple declaration; the second alerts the listener that the title in question is special above and beyond the mere description that follows.
Similarly, one could pronounce the article "thee" to emphasize that only one example of a thing exists: "She was the reason I stayed in Chicago." (Meaning she was the single or entire reason the speaker stayed in Chicago.)
In both examples, the difference is one of dramatic emphasis.
In your example above, I would pronounce Tolkien's trilogy "The Lord of the Rings" with no "thee" at all. To pronounce the long "e" in the first word of the title would be to suggest that there are pretenders to the title of Lord of the Rings and you want to make sure people understand that this one is the genuine article.