Tesla coil low pass filter are desirable to help isolate the radio frequencies of the primary tank from the power source. In neon transformer-based designs, the Low Pass filter helps to prevent destruction of the neon transformer's insulation. This is especially important as the primary capacitor in a tesla tank can resonate with the leakage inductance of the neon secondary.

The design of the low pass filter should be such that it passes kickbacks or voltage spike that travel from the primary tank circuit back to the power supply (neon transformer) to  RF ground. In other words, the low pass filter will allow low frequency signals (under 1000hz) to pass through the filter, however does not allow high frequency signals or voltage spikes to pass as passes these high Frequencies to RF ground. Tab on pictures to enlarge.