In some cases it is, and in some cases it is not. In New York City, for example, there are three major daily newspapers. While people will refer to them in conversation in ways that sound like "the New York Times", "the New York Daily News", and "the New York Post", in each case the number of words that form a true part of the newspaper's name (and thus the number that would be italicized) is different. In the first case, the newspaper's name includes the article and the city name, and so you would write "The New York Times". In the second case, neither the article nor the city name are actually part of the newpaper's name, and so you would write "the New York Daily News." In the case of the third newspaper, the official name of the publication has the city name, but no article, and so you should write "the New York Post." You would only know whether the article or the city name formed part of the actual name of the publication by looking at an actual copy of the paper.
GWB is right about citing newspapers in running text. However, in a citation in a bibliography, footnote, or endnote, you should include only the name as given on the newspaper's "masthead," the strip at the top of the first page, above the headline(s). So you would cite:
"WordReference Becomes World Authority on Language Usage," The New York Times, Feb. 17, 2009, p. 23.
"Ex-Cop Gives Word Usage Advice," New York Post, Feb. 17, 2009, p. 2.
"Boroughs Need Grammar Help," Daily News (New York), Feb. 17, 2009, p. 2.
"American English Continues Deterioration," The Times (London), Feb. 17, 2009, p. 1.
If the city name isn't part of the newspaper's name, you should include it in your citations unless the city of publication would be clear to your readers. For instance, if you were writing about something that happened in New York City, you could presume that your readers would understand that the Daily News you cited many times was the one published there. And some newspapers are "national" and not associated with a city, like USA Today in the United States. But lots of US cities have newspapers named "Times," "Post," and "Daily News" (or at least they did when even small cities had several newspapers), so you usually have to specify the city.
All citations are fictional, facetious, and intended for illustrative purposes only.