Tectonic and volcanic activity are caused by processes in the crust and mantle. Though the mantle very slowly flows and deforms like tar or putty, it is overwhelmingly solid (albeit very hot) rock. Magma exists only in certain regions of the crust and mantle, and even then mostly as a partial melt in a solid matrix, like water in a sponge, or slush.

The core does provide some heat from below, which helps power that activity, in addition to the heat in the mantle leftover form Earth's formation, and the heat in the mantle and crust generated by radioactive decay. But the core is not directly involved, and it being solid or liquid (it is both, actually, with a solid inner core and molten outer core) wouldn't necessarily preclude or permit volcanic or tectonic activity.