it is possible to give input that will cause an IIR filter to go into self-sustaining oscillations.
How could a stable IIR filter go into oscillation? And if it's unstable, I'd think most inputs would drive it into oscillation (but I haven't done much with IIR filters myself).
That's why FIR filters are much more common in control systems. Unconditional stability is a big plus. FIR filters also have linear phase response. However, the transport delays introduced by an FIR filter can be problematic, too.
FIR filters can also have non-linear phase response. A minimum-phase FIR filter has most of the energy at the beginning, just like an IIR filter. I'm assuming this is what you mean by "transport delay".
Why use an IIR filter? An IIR filter with a given number of taps can be made to have much steeper roll-off than an FIR filter with the same number of taps.
This is about the only reason, efficiency. Otherwise, a (much higher order) FIR can virtually match response, since one can always take the impulse response of the IIR, window it, and make an FIR out of that.