Interfacing VoltageREF IC with the ADC
Perusing the datasheets, the only place where the OPA365 is used to buffer the REF pin is in the ADS8319 datasheet. The OPA365 datasheet specifically requires a series resistance for any significant capacitance (see page 11 and figure 7 of the datasheet). This amplifier is stable for pure capacitive loads up to 1nF, a far cry from the 10 microfarad it is driving in this schematic.

I personally think this may be an error in the ADS8319 datasheet as the OPA365 device is not stable with such a large capacitance being directly driven; the 50 ohm feedback resistor may be there to form a compensation network, but given the variation of typical large capacitors (10% is standard) I would not trust the circuit to be unconditionally stable.

The reason for buffering the reference input could be due to the fact that the ADS8319 is a charge redistribution device (unlike the resistive types of yore) which leads to lower power at the expense of taking pulses of current during the charge stage of the conversion.

As the reference pin has to be bypassed by a large capacitor (acting as a reservoir for said pulses of current, which will be taken from both here and the input signal), you will need a setup that is stable; the diagram without the OPA365 achieves that (the reference you are using is stable with large capacitors provided you choose the correct device - see below).

So I would not use the OPA365 to buffer the REF input on the grounds that I could not guarantee the stability of the amplifier (which is at its most vulnerable in unity gain).

Note the output ESR requirement for the reference; it should be below 1.5 ohm (see page 10 of the REF505 datasheet) for stability. If you desire the lowest possible noise (probably not an issue in your application) then follow the guidance, but rather than relying on the capacitor ESR (where usually only the maximum is stated), specifically add a 1 ohm series resistor to the capacitor, which if ceramic, will yield the 1 ohm of output ESR.

Update. Added an option for those times when a distributed reference is required.

When using a distributed reference, it can be desirable to buffer the reference and decouple this buffered reference at multiple locations. For those scenarios, devices such as the LM8261 are excellent choices.

HTH