I've only really heard people discuss it in the context of the Fuzz Face design, but yeah, different battery types can have a variation in impedance, so in some kinds of circuits the battery type can have a small effect on tone. Different battery chemistry also effects the way a battery behaves when it discharges. Some fade away slowly, some drop off very sharply (Like NiCd's).
At the end of the day though I run all my pedals from a power supply. So much less drama.