When the inverter is on, it is "stronger" than the resistor, so there would always be current going through said resistor but T1OSC will be whatever output the inverter forces it at (think of the inverter as a very low resistance connected to the opposite voltage source).
This current going through the resistor, as well as the internal inverter circuitry, both cause a "power draw", this means that they consume energy over time. If you leave them on, a battery may last some amount of time. But turning them off means that they will not draw as much current (thus power, thus energy) and the battery will last longer. If this is not a battery powered application, you dont need to think about power draw.
Fun fact: by disabling some things you dont need at the moment in a microcontroller, it can go from draining/killing a battery in 3 days, to using that same battery for 15 whole years! Whenever you need the inverter, you turn it on, use it, turn it off and put the core to sleep.