Although the Earth's core does provide some heat for plate tectonics, one of the main sources of heat is not the core! The Earth's mantle and crust (and core to some degree) have small amounts of radioactive elements. These elements change to other elements, and when they do they produce heat that keeps the Earth's interior hot. If the Earth didn't have these radioactive elements then plate tectonics would have stopped by now.

If we suddenly cooled the Earth's liquid outer core then we would lose Earth's magnetic field. But I'm not sure if cooling the core would stop plate tectonics. My guess is that plate tectonics would start back up again because of the heat from the radioactive elements. Eventually though, enough radioactive elements will change (decay) to other elements that there won't be much heat to keep Earth's interior hot. Plate tectonics will stop then as the Earth cools.

If the earth's core cooled enough for the outer core to solidify, plate tectonics would stop, but also, the earth would stop producing its own magnetic field.

Earth's magnetic field is produced by metals circulating in the molten outer core. If the core solidified, the magnetic field would shut down, and the atmosphere would no longer be protected from cosmic radiation and solar wind. This could eventually cause Earth to lose its atmosphere. This is most likely what happened to Mars, which does not currently have plate tectonics or produce its own magnetic field but is thought to have had both in the past when it was hotter.

Since planets are very big and very hot (and produce heat inside from radioactivity), they cool very slowly, so we will (fortunately!) never see what happens for ourselves. Instead, we look for clues like the alignment of magnetic minerals in ocean crust, which tell us which way the magnetic field was oriented when that crust formed. This can tell us about how quickly the crust forms and how quickly plates move away from each other as well as how fast the magnetic field changes.

From this site:
1. "When the molten outer core cools and becomes solid, a very long time in the future, the Earth's magnetic field will disappear. ... This will make life on Earth very difficult for human beings and other life forms. When the Earth has cooled completely, the movement in the mantle will also stop eventually."

From this site:
2. "The forces that drive Plate Tectonics include: Convection in the Mantle (heat driven) Ridge push (gravitational force at the spreading ridges) Slab pull (gravitational force in subduction zones)..."

From this site:
3. "If the core were to cool completely, the planet would grow cold and dead. ... Cooling also could cost us the magnetic shield around the planet created by heat from the core. This shield protects Earth from cosmic radiation. The shield is created by a convection process caused by constantly moving iron."

If the earth's core cooled down, plates would stop moving and plate tectonics would stop.

This has happened to Mars. Mars' smaller size meant that its core could not contain as much heat as Earth's does, and now, the heat from the Martian core has escaped into space. So far as we can tell, Mars is geologically dead, or nearly so.