At maximum the current (let's say with a defective mosfet where the gate is shorted to source) can be:
Imax = 9 Volts / 150 Ohms = 0.06A
with the power dissipated on the resistor:
9 volts * 0.06 = 0.54W.
So if you have a defective mosfet yes the resistor will fry too. But then you probably have bigger problems
However, in normal functioning, the gate behaves like a capacitor of a few nanoFarads so the current
through the resistor will start at 0.06A but will decrease to nearly zero in a fraction of second.
In fact, after that moment, the current will be
9V / 15150 Ohm = 0.000594 A
and the power disipated completely negligible (as the 15k resistor is limiting the current to a very low value)
I'd say use whatever resistor you want for the 150.