The points made apply to Fourier analysis in general, not specifically to the FFT, and amount to asking whether the model you are applying is appropriate for your data. That question can only be addressed at the level of theory and by empirical testing; either way, it needs to be addressed in the context of your problem.
The statement you quote is quite vague, and seems rather sweeping. Fourier analysis is applied all the time to systems which are only approximately linear and stationary - it's a question of whether the approximation is good enough to allow the results to be useful. Again, this is a matter for investigation in each case.
None of this says anything about the FFT. The FFT is an algorithm for computing the Discrete Fourier Transform (DFT), which is the appropriate version of the Fourier Transform for data that are of finite length and are regularly sampled. It always returns a very good estimate of the coefficients of the sine waves which, when summed, make up the original signal - in this sense it does not have limitations. It's an excellent tool, but it's just a tool and whether it's the right tool depends on understanding the problem at a higher level.
