How much can the FFT noise affect the timbre analysis of a harmonic sound? The timbre of an harmonic sound (e.g. the steady part of a violin note) is defined by the amplitude of each harmonic (including the fundamental). In order to compare the timbre of two instruments (e.g. two violins playing the G string), I could compare those amplitudes. If the instrument are similar, I suppose that there will be a little difference in the harmonic distribution. Now, is the FFT a reliable tool for this kind of analysis? Should I take into account the noise that it introduces, or is it negligible?
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.)