From grammatical considerations, some thumb rules for convenience could be applied. However, the question is largely one of style and less about grammar.
Firstly, when the title begins with an article, it is not always necessary to cite the article of the title. 'The New York Times' is referred to as the 'New York Times' in practice.
I went to the The New York Times office this morning.
Secondly, when the title beginning article is the same as the article required before the title, it need not be repeated.
I went to the The New York Times office this morning.
Thumb rule (2) provides an alternative to (1) so you may chose between capitalizing the article and not.
In the case of different articles as in
I am reading a The New York Times article.
use thumb rule (1). Here the grammatical requirement of an article is more important.
It's actually NOT grammatical -- to the best of my knowledge -- to use "the" in the case of a pronoun. But even if it is actually grammatical, it is a very unusual construction, intended to give emphasis. In both cases.
The use of "the" in the case of a proper name is to indicate that one is speaking of the well-known person by that name, as opposed to another person who is named similarly, but who is not the famous one.
If my name were Bill Maher (and thank goodness it is not), and someone was introduced to me, they might ask (if they didn't know what the famous Bill Maher looked like):
"Are you THE Bill Maher?"
In almost all cases with using "the" with a proper name, it is pronounced "thee" as opposed to "thuh", and with some emphasis.
Since I speak German, I happen to know that in casual conversation it is fairly common for someone to speak of another by name using the definite article. A friend of a man named Otto might speak of him to another friend as "der Otto". This is normally not done in English, however, except in the case I mentioned.
Best Answer
House style prevails. How does the organization refer to itself? Do they provide identity guidelines for third party media? Nearly all newspapers take the, but there are as many exceptions as there are rules when it comes to applying articles to named entities.
Certainly, if it is part of the official name, it would always be included except where attributive: The New York Times, The Telegraph; but New York Times columnist Thomas Friedman and Telegraph motorcycle correspondent Kevin Ash.
In other cases, the article is not part of the official name, but it gets attached idiomatically: She writes for the Chicago Tribune, I subscribe to the South China Morning Post. Perhaps this is because many publication names are based around a common noun, and we naturally add the to most such names: The Wall Street Journal, not merely any Wall Street journal, same as the Royal Society, the University of Virginia, the Brisbane Lions, or the Socialist Republic of Vietnam.
Newspaper names not based on this formula (the Foo Newspapertype) do not accept articles: Barron's, never the Barron's, as with Roll Call, USA Today, Stars and Stripes, and so on.
But then there's Sporting News, never the Sporting News, because that is their house style. And JAMA is always simply JAMA, even though the full, official name is JAMA: The Journal of the American Medical Association.
See also