How close to Earth's core can organisms live?
There's a lot we don't know about life in deep caves, but we can bound the deepest living organism to at least 3.5 kilometers down, and probably not more than 30 kilometers down.

The worms recovered from deep mining boreholes are not particularly specifically adapted to live that far down: they have similar oxygen/temperature requirements as surface nematodes. 

The Tau Tona mine is about 3.5 kilometers deep and about 60˚ C at the bottom. Hydrothermal vent life does just fine up to about 80˚C, and the crust gets warmer at "about" 25˚C per kilometer. It's entirely reasonable to expect life to about 5 kilometers down, but further than that is speculation.

Increasing pressure helps to stabilize biological molecules that would otherwise disintegrate at those temperatures, so it's not impossible there could be life even deeper. It may even be likely, given that the Tau Tona life breathes oxygen.

I am certain no life we might recognize as life exists in the upper mantle.