does audio feedback consist of harmonic content?
The contradicting statements may be a result of looking at two different aspects of the situation.  As Hilmar pointed out in an earlier answer, any clipping (be it electronic or mechanical) will indeed produce harmonics.

However, feedback distortion (not necessarily reaching to the point of the loud feedback oscillation being produced) definitely does not consist of harmonics, or even any spurious frequencies added!  Feedback distortion comes from a frequency response with a very weird structure (very unnatural to the ear, often described as adding a metallic quality to the original sound).

The reason is the following: since the sound takes a fixed amount of time to travel from the speaker to the microphones (simplified analysis assuming single speaker and single microphone, no echos/reflections, and no attenuation of sound w.r.t. distance traveled), then the frequencies at which the feedback is positive, thus producing peaks in the frequency response, are the frequencies $f$ such that $$d = k \frac{\,c\,}{f}$$ where $d$ is the distance between the speaker and the microphone, $c$ is the speed of sound, and $k$ is a positive integer.  In other words, positive feedback (and thus larger gain) occurs at frequencies where the distance between the speaker and microphone is an exact multiple of the corresponding wavelength.

Now, you see, that means positive feedback occurs at frequencies $$f_k = k \frac{\,c\,}{d}~~~~k = 1, 2, 3, \cdots$$

There, you can see that the distortion is not harmonic in nature --- first of all, the above, assuming gains are low enough that you don't produce clipping or oscillation, is linear distortion (strangely enough, seeing how the feedback distortion is a very unnatural-sounding artifact); and second, the spectral locations of the "spurious" frequencies are fixed and equi-spaced, independent of the input frequency. In fact, harmonic distortion applies to an input tone (a sinusoid at a fixed frequency), whereas feedback distortion simply introduces a "weird/unnatural" frequency response, so it applies to signals with a spectrum covering ranges of frequencies.

A feedback oscillation may be subject to a combination of both effects: it may still carry a perfectly audible "feedback distortion" quality (especially if the gain is just enough to produce the oscillation), while it may or may not produce clipping that adds harmonic distortion to the mix.