A reader writes:
I'm a Caucasian male, married to a Chinese woman, who has four biracial children, and the whole family finds A Christmas Story hilarious - the "fa-ra-ra-ra" scene in particular. My wife grew up in China, and thus was not teased as a child for a foreign accent - it sounds like your correspondent (who is offended by the scene) grew up in a mostly-white US town where anti-Asian prejudice was frequently and openly exhibited. Different experiences cause different people to react differently to things - something to keep in mind.
That said, the speech impediment portrayed in the movie - substitution of "l" with "r" - is not typical of Cantonese native speakers (such as most Chinese immigrants to the US prior to World War II; the time period portrayed in the film).
The Cantonese language has an "l" sound but no "r" sound; it's more common for native Cantonese speakers who learn English to speak like Elmer Fudd - replacing "r" with "w", rather than inserting "r". Fa-ra-ra-ra-ra is probably more likely to be heard from a native Japanese speaker than a native Chinese one; as Japanese has a sound approximating "r" but none approximating "l".
And finally, no discussion of Asian stereotypes in American films would be complete without inclusion of Geddy Watanabe's hilarious portrayal of the Chinese exchange student Long Duk Dong in Sixteen Candles. Roger Ebert wrote of Watanabe's performance: "he elevates his role from a potentially offensive stereotype to high comedy". Compared to that, the "fa-ra-ra-ra" scene is tame stuff, indeed.