What are the critical characteristics of an op-amp for buffering reference voltages?

To summarize, you want to know what opamp characteristics you need to buffer a reference voltage.

The two most obvious are low offset voltage and unit-gain stability. Any offset voltage is a error directly added to the output. Without unity-gain stability, the thing will oscillate in voltage follower (unity gain) mode.

Other parameters depend on more specific circumstances. For example, will this thing be subjected to wide temperature swings. If so, low offset across a wide temperature range is important, not just offset at 25 °C or whatever.

Other issues come up depending on what supply voltages you have available relative to the reference voltage you want to buffer, and how stable those supplies are.

Do you expect the load to have a significant capacitive component? If so, you have to look at stability more carefully.

In any case, it would be a good idea to filter both supply inputs with a ferrite chip inductor or two followed by caps to ground. That way you don't have to count as much on the active power supply rejection capability at high frequencies.