Using a varactor diode to tune a resonance frequency
The circuit you're intending to use is not one of those commonly found in actual devices – exactly because the L has (nearly) zero DC resistance, so a lot of current will flow through it due to the DC bias.

It's important to notice that your (R||L||C) resonant circuit is not usually useful in this constellation in isolation. If you want to use it as the resonating part of an oscillator, you'd typically have some kind of amplifying element that feeds energy back into that oscillator; since we're talking AC here, the coupling can usually be done through a capacitor, and that means that you can isolate the biasing; you might want to think more in terms of:







simulate this circuit &ndash; Schematic created using CircuitLab

More common, however, is the configuration where the bias voltage is applied over two antiparallel diodes (from German wikipedia):



Notice the choke in the bias voltage path – since the ideal voltage source has zero impedance, it would completely "swallow" any energy stuffed into your oscillator, and Lb avoids that.