Although what @Fat32 wrote is correct, I think the potential instability of IIR filters is not the main reason for the instability of an adaptive IIR filter. After all, we can calculate the poles in each iteration and put a hard constraint to avoid poles out of the unit circle.
Even with an unconditionally stable IIR filter, we can end up with an unstable loop if the loop gain at certain frequency is large enough.
With FIR filters we are essentially solving iteratively a convex optimization problem without local minimums. The error term is linearly related to filter weight, this makes the problem both 
1-Well behaved=> So that you can easily converge
2-Easy to analyze => So that you can find a scaling factor that leads to the fastest convergence.

With the IIR filters, the problem is not convex and is nonlinear. If you look at the following block [diagram][1] you may think that it is linear with respect to filter coefficients. However, you can inspect that the input to $B(z)$ block contains the coefficients from $A(z)$ and previous iterations of $B(z)$.

[![Adaptive IIR filter diagram][2]][2]


 [1]: http://read.pudn.com/downloads74/ebook/269736/Adaptive%20IIR%20Filtering.pdf
 [2]: https://i.sstatic.net/BdHxp.png