Q. Would it be “the Color Purple musical” or “The Color Purple musical”?
A. The musical version of The Color Purple would be referred to as “the Color Purple musical”—where “the” is part of the surrounding text (and the The in the title has been omitted). A “the” belonging to the text could also be used before a title that doesn’t include an initial The. For example, a musical version of Star Trek might be referred to as “the Star Trek musical.”
Or consider other scenarios where a title that does include an initial The is used attributively (i.e., modifies another noun—like musical in the examples above). If you were to retain the The in the following example (where the title modifies character), the result would be clearly awkward:
Which Great Gatsby character do you dislike most?
not
Which The Great Gatsby character do you dislike most?
There’s no “the” at all in the first version of that example—which would also be true if you were to refer to “a Color Purple musical,” where the indefinite article “a” displaces the definite article The in the title. In general, when the title of a work is used attributively, be prepared to omit an initial The in favor of the surrounding text. See also CMOS 8.169.