I just came across the following sentence;
Many Americans love to watch reruns of the I Love Lucy show.
and think that the in there is extraneous. What rule applies to it?
It's definitely not extraneous. In fact, I'd say it was required.
Interestingly enough, if we remove the word show, then we would omit the article, too:
Many Americans love to watch reruns of I Love Lucy.
but we could not say:
Many Americans love to watch reruns of show.
unless the word was pluralized:
Many Americans love to watch reruns of shows.
The EF website lists 15 reasons when we use a definite article, but I'm having trouble figuring out which of those rules applies to this sentence. However, the CUNY School of Law has a webpage about the definite article with a rule that seems to apply:
In general, the article THE is used when a noun has a specific reference to something unique.
In this case, the noun is show, and it's referring to a particular (i.e. a unique) show.
On the EF website, that rule is expressed as:
Use the in sentences or clauses where you define or identify a particular person or object.
I think those are essentially two ways of trying to say the same thing, but the wording of the former maps a little better to the sentence in this question. An example on the law school website is:
which aligns pretty closely with: