Does the inverse Fourier transform only produce stationary time signals?
The FT of a finite length sine wave burst consists of more than just the one frequency of that sine wave.  A finite width time domain signal has an infinite width FT.  Same for an FFT, a rectangular window produces a full width periodic Sinc in the FFT result.  So even if the phase of the 1 sine wave result bin stays constant, the phase of all the other frequencies will rotate as the position of the window changes.  Two different rectangular windows will NOT map to the same FFT, but convolve against any single sine wave differently.

Therefore an inverse FT or IFFT of the full spectrum can produce a signal whose statistics can vary significantly over different portions of the result, by reproducing a window which can be zero in parts of the result, and non zero in others.