Fast MOSFET, how to choose resistor values and how this influence speed
Think of the MOSFET gate as a capacitor. To turn the FET on, you need to charge this capacitor to about 2 V; higher voltages will further reduce the on-resistance of the FET. 

The R is used to reduce the rate of charging of this capacitor. When you drive in steady state (i.e. after a 'long' time), the capacitor is charged to the driving voltage, and there is no current in the resistor. 

Generally power dissipation in this R is not a critical issue -- it would only be an issue if you were driving FETs at 10's of MHz.

In some cases, especially when the wire between the driver and the FET is long, a smaller value (100 ohm) is used to prevent the long gate lead from picking up RF signals or feedback from the drain and triggering a high frequency oscillation.