What opamp characteristics for voltage reference
Think about what you need and what you don't need.  Then think how those needs are effected by your design parameters, perhaps they might be ultra low supply current or a massive operational temperature range.

Once you decide what matters, then find how these parameters change over time and temperature.

It is impossible for any additional components not to affect the performance of any system in some way, after all you have more variables. Whether these affects are important or not is another thing.

For a reference what you need a stable, low noise voltage over it's operating range.

What you don't need is some circuit that adds an error or increases the noise.

The op-amps AC parameters are almost useless here, but the DC parameters are really important.

It is not easy to give a concise list of parameter names as different data-sheets often describe the same things differently (if at all), though here is a list of things to consider:
Some op-amps can have a significant input offset current and input offset voltage.  How is this current going to affect the reference? Is the offset going to add to your overall error.
They might also have some sort of output offset (this is often trimmed)
They have a PSRR, power supply rejection ratio, this is a measure of how the power supply noise is passed to the output.
Is the output voltage capable of achieving the reference you need?
Is the input capable of being driven to the input reference you need?