Yes, although you can't completely eliminate the drain on the battery. You need some current flowing through the plate and foot. Here is a circuit that has a very low current consumption in the OFF state.
This circuit uses two transistors connected as a Darlington pair. This configuration has very high current gain - the current gain is the product of the current gains of the individual transistors. So it needs very little base current to turn the transistors ON and illuminate the LED.
You can make a Darlington pair with two separate transistors, as I have shown in the diagram, or you can buy a Darlington pair in a single package - for example, the MPSA14.
This base current comes from R1, a 1 megohm resistor. When the contacts are closed, the bottom end of R1 is held at 0V and there is no voltage available to bias the transistors into conduction. In this state the current drain is 1 µA multiplied by the battery voltage in volts; for a 9V battery, this will be 9 µA (microamps). This is a pretty low operating current and a standard PP3-type 9V battery should last pretty much its shelf life.
When the contacts are broken, R1 pulls the voltage at R2 up towards the positive supply and this causes Q1 and Q2 to conduct, illuminating the LED.
Q1 and Q2 do not turn on "hard", i.e. they don't saturate, so there will be some voltage lost across Q2. There is also about 2V dropped across a typical red LED. The value of R3 that I've chosen gives an LED current of about 10 mA from a 9V battery.
If you want to learn more about this circuit, and basic electronics in general, Google Darlington pair and Ohm's Law.