The title of a composition is self-contained; an article (a, an, or the) appearing as the first word of the title cannot serve that role as well as function as an article preceding the title. Discussions and revisions follow each example of this principle below.
1. The Apprentice guru has seen four of his casinos go bankrupt.
This sentence cannot appropriate the first word of the title of the television program to serve as the sentence’s opening article; therefore, the sentence lacks an article. To resolve this issue, use a workaround convention—insert an article for the sentence and elide the title’s article: “The Apprentice guru has seen four of his casinos go bankrupt.” (Essentially, unitalicize The, but understand why you did so.) Alternatively, relax the sentence by inserting the article, relocating the noun that the program title modifies so that it precedes the title, and inserting of after that: “The guru of The Apprentice has seen four of his casinos go bankrupt.”
2. The Danish Girl star showed up in court in Los Angeles on Friday with a bruise on her face.
Use the same solution here: “The Danish Girl star showed up in court in Los Angeles on Friday with a bruise on her face.” (Or write “The star of The Danish Girl showed up in court in Los Angeles on Friday with a bruise on her face.”)
3. Smith is expected to shoot The Untouchables remake.
The same problem exists, and the same solutions apply, when the title appears elsewhere in the sentence: “Smith is expected to shoot the Untouchables remake” (but, in this case, lowercase the in addition to unitalicizing it) or “Smith is expected to shoot the remake of The Untouchables.”
4. Don’t miss the A Christmas Story marathon.
Titles beginning with the article a (or an) should be treated the same way: “Don’t miss the Christmas Story marathon” or—with further revision necessary in this case—“Don’t miss the marathon movie event celebrating A Christmas Story.”
5. Berrigan credited Dorothy Day, founder of The Catholic Worker newspaper, with introducing him to the pacifist movement and influencing his thinking about war.
An article that begins the title of a periodical publication should never be italicized: “Berrigan credited Dorothy Day, founder of the Catholic Worker newspaper, with introducing him to the pacifist movement and influencing his thinking about war.” (This is a style convention of necessity, because periodicals are inconsistent about whether they use an article—for example, compare copies of two preeminent American newspapers to note the difference in the official titles of the New York Times and the Los Angeles Times—and it’s a burden to try to keep track of which publications follow which style.)
Would it actually be wrong, or simply awkward, to refer to “the A Christmas Story marathon”? It’s stylistically clumsy and easily avoided, but I would have trouble specifying why it would be technically wrong. Just curious.
I think you meant to capitalize the The in The New York Times to make the point in your last sentence:
“… to note the difference in the official titles of [T]he New York Times and the Los Angeles Times—and it’s a burden to try to keep track of which publications follow which style.)”
I would consider using a possessive. Proofreadnow.com offers this example:
When a proper name is in italic type, its possessive ending is preferably set in roman:
The National Review’s fortieth year of publication
In this example (unable to reproduce it here—Ctl-I doesn’t work), “National Review” is italicized, while the apostrophe and “s” are not. One could, at least, apply this to something like “The Danish Girl star.”
I’m not entirely convinced that the possessive isn’t a touch more grammatical, too.
I don’t see the issue in four of the sentences. Capitalizing and/or italicizing the article doesn’t impact how it is read. You’re basically saying that one grammar rule related to articles is more important than another grammar rule related to proper titles.