You like to take a long view, don't you? If the amount of heat in the depths of our planet had to be removed by our efforts, you would be thinking in terms of tens or hundreds of millions of years. However, that is not the whole story. As Kelvin did not realise when he calculated how long it would take for the planet to cool down, that heat is being replenished from radioactive elements down there, or it would have cooled down long ago.

That said, if we nonetheless did manage to cool the mantle and core beneath our feet, it would put an end to plate tectonics (continental drift, earthquakes, orogeny, monsoons, etc) the planet would develop a flatter surface, covered  with a slightly shallower ocean.

Probably the icecaps would disappear too, and everyone would blame the darkening of the polar bears on global warming.

But as I indicated, don't bother to hold your breath...

First a bit of pedantry, extracting all the heat from anything is, as far as I know, impossible. So rephrasing the question to What would happen if the earths core were cooled to the temperature of space? (about 2.7K)

I would guess that the core would contract with the loss of thermal energy and would form one large crystaline mass with maybe only a small magnetic field. The reduction in space taken up would leave the crust unsupported and it would wrinkle. Also due to the conservation of momentum the rate of rotation would also increase shortening the length of a day.

If the core was the only part cooled and the mantle remained the same temperature then the convection currents would be split with currents going from the hottest part of the mantle to the surface as well as from the hottest part to the core.