What's the purpose of the OpAmp in this circuit?
The zener and 120 ohm resistor form one arm of a bridge. The 30K resistor and BJT form the other arm. The bridge will be balanced when the transistor \$V_{CE}\$ is the same as the \$V_{BE}\$. The op-amp forces said bridge into balance. 

Under balance conditions, the current through the zener is about 5mA (held constant). The transistor \$I_C\$ is about 220uA. 

The output voltage is the zener voltage plus \$V_{BE}\$. At the voltage the manufacturer has chosen for the buried zener (7.2V - \$V_{BE}\$), the positive temperature coefficient of the zener matches the negative temperature coefficient of \$V_{BE}\$ , so the overall temperature coefficient is close to zero and the output voltage is nominally 7.2V. 

In the old days (before high-performance IC references became common), one of the best references you could buy was the 1N821A (and its cousins), which have a diode internally in series with a zener (you can tell this indirectly from the data sheet because the 'reverse' current is blocked by the diode to a maximum of a few uA). 

Of course in the venerable LTZ1000, the tempco is usually further improved by stabilizing the die temperature at a temperature above ambient using the internal heater and temperature sensor.