How much can the FFT noise affect the timbre analysis of a harmonic sound?
An FFT is a finite length transform.  Therefore one does need to take into account any artifacts introduced by the window (implied rectangular, or optional Von Hann, et.al.).  

There is some controversy, but you might consider sampling using a greater ADC bit depth than 16 bits at a rate higher than 44.1 ksps to feed you FFTs, and use one of the typical double precision FFT libraries to reduce any numerical noise.  (...and how you mic the violin may make a far greater difference than any of the above.)