Effect of low pass filter on the time domain
If you sample a pure sinewave for an integer number of periods, the samples will sum to zero (it will have equal "ups" and "downs").  If you zero an FFT bin, that's the same as subtracting a sinewave of the same magnitude and phase as was in that bin, or identically, adding a pure sinewave of the same magnitude but opposite phase.  So zeroing bins is the same as adding sinewaves that are exactly periodic in FFT aperture (as those are the basis vectors of a DFT).  So zeroing any set of bins (other than the DC or 0 Hz bin) adds samples than sum to zero, thus having no effect on the total sum of the time domain samples after the modification.