What happens when the poles of this z-transform function are outside the ROC for a signal?
Each ROC has it’s own impulse response  .  There isn’t a general impulse response or a single ROC in most cases. 

Your problem can have 3 ROC with 3 impulse responses and 3 stability conditions 

One ROC is the causal ROC, Another is the anti-causal ( running the filter in reverse time) and the last, which isn’t common has a name that requires me to look it up, but this isn’t my homework problem so i’m going to stay put.  

I know that in the first edition of Oppenheim and Schaefer, they explicitly say you can’t write down an impulse response from the Z transform without stating the ROC