I'm not aware of any work on this topic in linguistic phonetics, but there may be something out there in musical acoustics for voice. The main problem for quantifying inharmonicity is detecting harmonics exactly, and the main issues can be seen using Praat. Frequency information comes in fixed-width bins which is inversely related to the length of the analyzed signal: the longer the analysis window, the smaller the frequency step between successive frequency computations. Each glottal pulse is fairly short, say in the range of 8 msc, and an analysis that looks at the harmonic shape of just one glottal cycle will have a much larger frequency step. A large frequency step means higher uncertainty as to the actual frequency which a reported peak corresponds to.
Comparing an 8 msc window to an 80 msc window, the shorter window has a frequency resolution of about 86 Hz whereas the longer window has a frequency resolution of about 11 Hz. In the former short window, it's not practical to try to identify harmonics. Even increasing the window length to 16 msc (about 2 glottal pulses), the identified "peaks" are way off from integer multiples of the fundamental (about twice the actual multiple). With an 80 msc window, frequency resolution is mathematically pretty decent, but even then you will get apparent inharmonicity, because of the limited frequency resolution of the apparent fundamental. If you look at a higher harmonic such as the 10th, the apparent peak should occur at 10 times the frequency of the fundamental, give or take one-tenth the frequency resolution (so about 1 Hz for an 80 msc window). The computed frequency of the fundamental, on the other hand, is that apparent peak, give or take 10 Hz. Autocorrelation methods for pitch extraction can solve that problem, as long as pitch changes negligibly within the window. What you will probably find is not that higher harmonics are "off" relative to the fundamental, rather, the harmonic peak of higher harmonics occurs a multiple of the fundamental, and the lowest peak that supposedly reflects the fundamental is off a bit.
With a piano string or tuning fork, you can strike a note, sample it and get a long analysis window with excellent frequency resolution (assuming that you don't have gremlins jumping up and down on the vibrating string). Human speech is full of acoustic gremlins, so the longer the window of analysis, the more "noise" you're going to introduce (especially variation in the shape and duration of individual glottal pulses). This may be more controllable with professional musicians performing sustained notes, but that is not speech. Since perfect vs. imperfect harmonic series is not linguistically salient, I think you won't find any linguistic phonetic literature on this. You may find something in the realm of speech pathology and dysphonia.
One way to overcome the frequency-bin problem is to (very carefully) select a small number of cycles – two – selecting the zero-crossing at the same point of two successive cycles, then concatenate a number of copies. You will get spurious harmonics (every even harmonic, I think, will be a lower-amplitude artifact) and also you have to define "peak" in a special way (it should be the midpoint in the series of "peak-ish" numbers, but that point may be a negative number with the set of dB values). But, the frequency resolution will be improved to the point that if F0 is 110, you'll get harmonics at every integer multiple of that. This minimizes the changes over time that characterize speech and introduces fewer artifacts (esp. compared to iterating a single cycle).
I don't know of a published source on this, but experientially, I know that synthesized complex waves built from sine wave components of integer multiple periods sound mind of mechanical. A bit of variation in the components (within a cycle and between cycles) makes it sound somewhat more natural. I expect that too much variation has the opposite effect.
