What does the uncertainty principle say about recursive filters?
The uncertainty principle basically says the same thing for IIR filters: "the steeper the filter, the longer the Impulse Response". The only difference between FIR and IIR is that you have to use a somewhat different definition of "filter length".

While all IIR filters are technically infinite, they have very different decay rates. You can define a quantity that basically describes "energy decay over time". For most IIR filters this is an exponential decay (often dominated by the pole-pair closest to the unit circle) and the decay rate is directly related to the filter bandwidth. 

An interesting quantitative approach to determining decay rates can be found in the area of reverb time calculation, using "time reversed impulse response integration".