At what point does plate tectonics stop?
One permanent threat to plate tectonics is the oceans vanishing. The scientific jury may still be out on this matter, but most geologists and geophysicists consider water to be the lubricant that makes plate tectonics possible. In a billion years or so, the Sun will have become 10% more luminous. This is conjectured to make the Earth to undergo an unstoppable moist greenhouse / runaway greenhouse, and the oceans will vanish.

The core is still emitting residual heat from the formation of the Earth. As the core cools, iron in the outer core freezes onto the inner core. This is freezing is an additional source of heat. The solidification of the Earth's core thus represents another permanent threat to plate tectonics. The liquid outer core is conjectured to have frozen solid less than three billion years from now.

A nearer term threat is the formation of the next supercontinent. Plate tectonics may operate in fits and starts. What makes subduction zones form is not known. What is known is that a major subduction zone vanished when India collided with Asia, and no new subduction zones formed elsewhere to take its place. If this conjecture is true, plate tectonics will temporarily stop in a few hundred million years when the next supercontinent forms, only to restart later when too much heat stress builds up inside the planet.

This is all highly conjectural. No scientist will live to see their conjectures falsified.



References:

Kasting (1988), "Runaway and moist greenhouse atmospheres and the evolution of Earth and Venus," Icarus 74.3 : 472-494

McDonough (2003), "Compositional model for the Earth's core." In Treatise on geochemistry 2 547-568.

Silver &amp; Behn (2008), "Intermittent plate tectonics?" Science 319.5859.