Segregation in the South didn't make many provisions for grades colour; consequently, Asian-Americans (as well as Native Americans) occupied an uncertain place in between the two. As such, the exact answer depends on the time and precise place.
This uncertainty is neatly demonstrated by the testimony of Mary Tsukamoto, who had been interned in Arkansas during the Second World War - itself a racist policy not inflicted on whites (though not a product of Southern segregation). Yet, Arkansas bus drivers did not treat her as "colored" either:
We could not believe the bus driver's tone of voice as he ordered black passengers to stand at the back of the bus, even though there were many unoccupied seats in the front. We wondered what he would do with us, but he smiled and told us to sit in the seat behind him. We were relieved but had strange feelings; apparently we were not "colored."
Tsukamoto, Mary, and Elizabeth Pinkerton. We the People: a Story of Internment in America. Laguna Publishers, 1988.
On the other hand, the legal ambiguity meant that the status of Asian-Americans was in flux, and could theoretically be modified. In fact, some communities succeeded in raising their social status from "coloured" to "white". Loewen's highly influential 1971 study on the Chinese community in Mississippi noted that, over the space of one generation:
The white man's evaluation had to be taken seriously, because he controlled the distribution of such valued commodities as public accommodations, legal treatment, and public education. [As a result the Chinese] began to take seriously white Mississippi's low placement of them. They then worked systematically to eliminate the causes of that treatment, in order to rise from Negro to white status . . . Their transition was a racial one. That is, the Chinese entered white society as a group, a racial group; individuals were not accepted separately.
Loewen, James W. The Mississippi Chinese: Between Black and White. Harvard University Press, 1971.
In fact, by 1967, the Chinese were literally card carrying "white" people according to the "W" on their driver's licenses. Of course, this was not achieved easily; Loewen notes that on every step of their way - from employment to education to membership in organisations - the Chinese met with varying levels of white resistance.
As for movie theatres, Choong Soon Kim - an anthropologist of Korean descent who worked on an reservation in the South - provides an interesting view:
This is the case in Kim's refusal to enter a segregated theater, particularly after he finds that, although his presence in public facilities in the South undergoes discomfiting scrutiny, it has not heretofore been challenged. . . . he does not want to be humiliated in front of his students and suffer a loss of authority.
Bow, Leslie. Partly Colored: Asian Americans and Racial Anomaly in the Segregated South. NYU Press, 2010.