Could mitral valve regurgitation cause anemia?

Unlikely. If one has mixed mitral valve disease with a heavily calcified mitral valve that is both stenotic and regurgitant, one could imagine marked turbulance of the blood flow across the mitral valve with destruction of rbc's however, this is very unlikely.
Unlikely. Unless this is after a mitral valve replacement and there is significant peri-valvular leak causing rbc destruction.