Accuracy of time domain signal when one frequency is absent
When a signal is expressed as a linear superposition of orthogonal basis functions, like a DFT, when you remove one orthogonal component, that error is orthogonal to the remaining terms, so any perturbations to those remaining terms will not have an effect on the error due to the removal.

Nevertheless, missing "data" is an active area of research.  If you can make some assumptions about what is missing, you can sometimes, with varying success recover that data.

There is a very popular family of algorithms known as the "EM algorithm"

Another more recent area is known as "Matrix Completion"

There is a mathematical property called "restricted isometry" that is used in sparse sensing that may of use in your problem.

When you google scholar the term "incomplete spectrum", you get hits like:


  E. Serpedin, "Subsequence based recovery of missing samples in
  oversampled bandlimited signals," in IEEE Transactions on Signal
  Processing, vol. 48, no. 2, pp. 580-583, Feb 2000. doi:
  10.1109/78.823989 Abstract: A new approach for recovering an arbitrary finite number of missing samples in an oversampled bandlimited signal
  is presented. This correspondence also proposes an approach for
  recovering the original signal's spectrum from the spectra of a
  certain number of subsequences, obtained by downsampling the original
  sequence. Closed-form expressions for the missing samples in terms of
  the known samples are obtained by exploiting the linear dependence
  relationship among the spectra of downsampled subsequences keywords:
  {bandlimited signals;signal reconstruction;signal sampling;spectral
  analysis;closed-form expressions;downsampled
  subsequences;downsampling;known samples;linear dependence;missing
  samples;oversampled bandlimited signals;spectrum;subsequence based
  recovery;Bandwidth;Closed-form solution;Equations;Fourier
  transforms;Frequency domain analysis;Iterative algorithms;Iterative
  methods;Numerical stability;Sampling methods;Signal restoration}, URL:
  http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&amp;arnumber=823989&amp;isnumber=17842


If you can express your problem in terms of any of these paradigms, you can arrive at solutions, and some have bounds on errors.

As was suggested in the question, an often useful simple heuristic is to interpolate the missing data. The most straightforward way to get an handle on error, is trial and error on actual data.  A lot of times, acts of technical desperation work out.