Very narrow-band IIR-filters are highly unstable!

They either are, or they are not (Sorry for the nitpick. That's semesters of system theory related classes speaking).

It is, however, very easy to make them unstable.

Even the least rounding error can make the whole thing to oscillate.

This isn't really a big concern with the Goertzel algorithm, because the resulting filter is only marginally stable in the first place (its zeroes and poles lie on the unit circle in the z plane) and therefore it shouldn't be run indefinitely anyway. You run it for a number of samples, check if they contain the frequency in question, and reset the filter.

Actually, by modifying the filter a little you can make its output equal to what the DFT result at the frequency in question would be - without having to keep all the samples in memory. Definitely a good point in memory-constrained situations, or when you're only interested in a small number of frequencies instead of the whole spectrum.

http://en.wikipedia.org/wiki/Goertzel_algorithm

Also, IIR-filters do not support decimating.

On the other hand, they don't require keeping all the old samples in memory, and all the associated buffer management.