A solid "it depends". The overtones are all sinoid signals, and sinoids stay sinoids given linear shift-invariant systems. Which most elements of sound transmission are. So if there is no significantly non-linear element after the sound-generating disharmonic element, the "unnatural" overtones are all there are.
With a piano, this will be mostly the case. If there is any snaring part in there, the overtones of the snare will be harmonic overtones clashing with the disharmonic ones.
As an extreme example, take a free reed instrument like accordion, harmonium, bandonion. The metal reed generating the sound is vibrating by bending, and there are several higher modes of bending and torsion that the reed can also exhibit apart from the fundamental. Those higher bending modes have frequencies wildly different from harmonics but due to the overall shape of the reed and the (destructive) manners of tuning it, sometimes they might come close to a harmonic in which case the reed tends to sound unclean and become more liable to breaking.
Now here the trick is that the mechanical oscillator has disharmonic modes of oscillation, but the actual sound is not generated by listening to the vibrations it may transfer to the instrument (which is rather less than more desired) but because the vibrating reed "punches holes into the air stream" which is quite a nonlinear operation and generates rich overtones. Those air stream overtones, in contrast to the vibration overtones, are a function of the "form of the holes" punched into the air stream and thus are strictly harmonical as they are part of a periodical signal with the period of the fundamental.
Another example is an electric guitar: both the magnetic pickup process as well as any additional distortions added by effects and amplifier are nonlinear and will tend to generate harmonic overtones while the vibration of the thicker strings might be responsible for disharmonic overtones. An acoustical guitar in good shape, in contrast, will almost only deliver the (slightly disharmonic) overtones from the strings without adding competing harmonic ones of its own.