Broadly speaking, the robustness of the superconducting state depends on the specific material, its pairing mechanism (for Cooper pair creation), type of impurity (e.g. chemical doping, site disorder, magnetic ions, etc.).

I won't comment on specifics (and take some of this with a grain of salt) bc I'm a bit out-of-date and don't work on superconductivity directly, but I have sat through many talks and work on electronic transport phenomena. For conventional BCS superconductors (SC), the superconducting state can be fairly robust to impurities as long as they are not magnetic, although I believe they tend to decrease the superconducting transition temperature (T_c).

With unconventional SC, the phase diagram is a bit more complex due to pairing mechansims that are still an area of research. In the superconducting cuprates, the archetypal phase diagram is a superconducting dome (or domes I hear...) as a function of hole-doping - this means that there exists a minimum amount of chemical impurity (substitutional doping) that creates a superconducting state, an optimal one with the highest T_c, and then a maximally allowed doping where competing states seem to interfere with the SC state. I believe this is a common occurrence in doping phase diagrams, although the interpretation of the underlying physics may differ: e.g. heavy fermion systems and quantum criticality, versus Fe-based SC, versus SrRuO-214. Once again tho, SC is a tangential topic for me, so please do give this a fact check. The idea here is that chemical impurities do not necessarily destroy the SC.

What i believe is true generally for the unconventional SC is the idea of competing ground states. Charge and magnetic order (both long-range and short-range) tend to surround SC areas in the phase diagram. Adding impurities then mess up this delicate balance and tip the system towards one type of order or another.

If you were to dig further into this vast topic, I'd probably suggest going the route of looking up 3 broad topics: 1) effect of chemical substitution on elemental superconductors (old and purity is suspect in some data but well-understood phonon-mediated models), 2) strontium doping in the lanthanum cuprates (La2-xSrxCuO4) and oxygen doping in La2CuO4+d (simplest cuprate systems but still rich in physics), and 3) effects on magnetic and transport properties in metallic alloys (to understand effect of impurities on the normal state of metals).