Any affect on the diff if a smaller diameter tire spare replaces a rear tire gone flat ?
If you have an open differential, no. IF you have a LSD, perhaps. Either way, I'd get the tire fixed ASAP. This is not meant to be a forever thing. This gets me to the next exit if there's a bad problem.
As Stan said, a smaller, emergency spare (or space saver) spare with an open differiential will work. You don't want to be driving it this way a long time (the space saver won't take the cornering loads the full size tire on the other side will) so get the tire fixed right away- that's kind of a given. An LSD is designed to have the same size tire on each side; I would think you wouldn't want to run it for even a short time with different diameter tires. The friction plates would be continually moving at different speeds, creating heat. What to do- I'll ask my friend Bruce next time I see him...
@Stan Galat also said- "I've got a 3" wide AL drag wheel and a space-save spare. It's pretty ridiculous, but it'd work in a pinch."
Remember guys, it's for emergencies only; hopefully you never have to use it.