How big does a pointed piece of tungsten need to be to self-propel into the earths core by gravity alone?
It will never get past the mantle. 

The tungsten (or tungsten carbide) will chemically react with the rocks surrounding it to dissolve and form other minerals (silicates), or even worse: get diluted to trace levels in mantle minerals. Compare this with dropping some grains of salt in the ocean and asking when they will arrive to the sea floor. Your block
of tungsten will probably need to be on the scale of tens of metres or more (km?) to escape that.

Furthermore, the mantle is convecting so it will keep your thing from sinking directly down. Compare with leaves falling from a tree in a thunderstorm. This will encourage the chemical mixing I mentioned before.