Not necessarily an offensive production, but along the same lines as the
Avenue Q casting in OP's post: I once saw a community theater production of
A Chorus Line, and the part of Connie Wong was played by a white actress. That would've been fine if they couldn't find an Asian actress, but one of the dancers who was cut at the very beginning was Asian. Her nondescript character is the first to sing ("I really need this job / Please, God I need this job / I've got to get this job!") and the Asian chick had a pretty good voice, judging by that bit, and she was petite. I kept wondering why she wasn't cast as Connie. It was weird.
Salve, Regina, Mater misericordiae
Vita, dulcedo, et spes nostra
Salve, Salve Regina
Ad te clamamus exsules filii Eva
Ad te suspiramus, gementes et flentes
O clemens O pia