What is the purpose of a diode and resistor at the gate of a FET in this LED driver circuit?
It is normal to drive the gate of a MOSFET through a resistor (normally under 100R). It is also seen a diode exactly as connected in the schematic.

The reason is that the gate is a capacitive load to the driver. This means that is has to charge when turned on, and discharged when turned off. If there is no resistor, the driver has to be capable of providing the peak current to charge the capacitor. On the other hand, to turn off the MOSFET you want it to discharge as fast as possible (for a fast turn off), and not through a gate resistor. 

Obviously if you short circuit the resistor, there is no need for the diode. I'd say that in this case it's a reserve in the PCB to allow for MOSFETS with higher capacitance in the gate.