Designing an IIR to generate a specific data sequence I know that it's possible to design an IIR with specific poles and zeroes to create specific frequency responses.

Is it also possible to design an IIR such that when you give it an impulse, it generates values of a desired sequence? Or at least a sequence within some tolerance of those values?
Yes, there are time-domain design methods for IIR filters. One of the best-known ones is Prony's method. It is well described in the book Digital Filter Design by T.W. Parks and C.S.Burrus (ch. 7.5).

If the desired sequence is (right-sided) periodic, then there's a very simple and exact solution possible (which is usually an exercise for DSP students).