The method used for the FFT utilizes that fact that the time series has equally spaced data points. When some data points are absent, masked, or otherwise invalid, then the method may produce invalid results. This situation is commonly referred to as the missing value problem and is not infrequently encountered. There are numerous methods for handling this problem and details of several methods are included in section 13.8 Spectral Analysis of Unevenly Spaced Data on pages 575-584 in the reference below. Numerous other methods appear on the WEB.
Reference: "Numerical Recipes in C: The Art of Scientific Computing" by William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling . Publisher: Cambridge University Press 2 edition (October 30, 1992)