#acting
I don't think there is a hard and fast answer to this question. A lot depends upon the nature of the work and the intentions of the people (producers/directors/etc.) casting the actors.
For example, the works of Shakespeare have become so widely distributed in Western/European culture the ethnicity of the characters is not strictly relevant. If Hamlet is played by a white actor, a black actor, or an Elcor actor, it doesn't really matter.
On the other hand, would you have Hulk Hogan star in The Stephen Hawking Story? How about Peter Dinklage in 20,000 Women: The Story of Wilt Chamberlain? Keanu Reeves in anything but "Bill & Ted" or "Speed"? At some point, the mismatch between actor and role is large enough to damage the credibility of the work.
The tricky borderline happens when the ethnicity of the characters being portrayed is germaine to the story being told. Arguably, only Danes should be cast for roles in Hamlet, but as I noted above, the nature of the work and the universal themes presented suggest the Danish-ness of the characters is not particularly important. On the other hand, in a production of Roots, the fact people of one ethnic background are the enslaved, while people of another background are the enslavers is highly central to the nature of the work. You might cast red-haired pale-skinned Irish people as the slave characters, but that's a fundamental change to the work. Maybe there's an artistic point to be made by doing so, but that makes the production a derivative work, not true to the original intent of the story.
I think the instances where this sort of thing causes the most uproar is when the artistic intent of the work DOES incorporate ethnic background as a significant part of the story, but that intent is ignored/subverted because of "box office" reasons.
Two works that come to mind recently are Aloha, where the majority status of Asian/Polynesian people in Hawaii is whitewashed away and a 1/4 Native Hawaiian-1/4 Chinese major character is portrayed by whiter-than-white actress Emma Stone, and the upcoming Ghost in the Shell, where Scarlett Johansson portrays an Asian character...because ScarJo is "box office" while Ming Na-Wen is not.
And, of course, the argument over this topic intersects with other issues about Mass Media Culture's obsession with youth and whiteness. If ScarJo plays the main role in Ghost in the Shell because she's already "box office", when does Ming Na-Wen get to play a role which brings her critical acclaim and name recognition, so that maybe Na-Wen is "box office" enough to portary Black Widow in the next Avengers reboot?
As with most topics this complex, the answer boils down to "it depends"...and the discussion starts from that point.