Why can adaptive IIR filters result in unstable solutions?
The IIR filter doesn't have to be unstable, but it has the potential of being so; unlike the FIR case which doesn't have even the potential.

One reason for the (potential) unstability of an IIR (adaptive) filter is the numerical issues due to coefficient quantization. When the poles are closer to unit circle this will be critical.
This is especially important if you are using coarse quantization (like in an legacy 8 bit system?) or you are forcing the limits of your numerical precision. 

Furthermore, during the adaptation process, the chaotic behaviour of the input (which is possibly nonstationary) may lead into unbounded updates of the coefficients. When the errors get quite large during a transient stage, the resulting update on the coefficients can also be quite large leading (depending on the adaptation algorithm) the filter to fall into an unstable region...