What properties must a rocky body possess in order to exhibit plate tectonics?
Plate tectonics appears to require multiple factors all arraigned in feedback loops.


The planet must be large enough that it's surface to volume ratios is low enough to trap enough heat from radioactive decay to power tectonic motion. Convection requires a minimal threshold of energy trapped in the material before that trapped energy causes mechanical motion. Planets smaller than earth radiate internal heat away to quickly because of their larger surface to volume ratios. Thus little convection and little tectonic motion.
The planet must have sufficient mass such that it's escape velocity exceeds that of a hydrogen ion ejected from a water molecule via UV hydrolysis. If the planet does have sufficient mass, then the hydrogen release by UV hydrolysis will be recaptured. If not, the hydrogen will escape the planet and its orbit and the planet will eventually desiccate. Earth has enough mass to retain hydrogen, Mars and Venus do not. In the past, Mars had water but it leaked hydrogen over the eons until it became a desert. Venus likely suffered the same fate. 
Oxygen producing life: Even with sufficient mass to trap hydrogen, a planet will still leak free hydrogen from having the hydrogen rise to the top of the atmosphere and there being knocked off by the solar wind. Free oxygen prevents this by a) forming ozone thus blocking the UV that breaks down water molecules and b) recombining with the free hydrogen that does form to produce water again. 
The presence of large amounts of water produce hydrated minerals which, on the size and time scales of tectonic motion, behave more like a deformable putty than solid rock. These deformable minerals allow tectonic motion. Without them, tectonic forces would cause shattering instead of drift. (See: Plate Tectonics and Water, and, Role of water in the tectonics of Earth and Venus.)
Tectonic subduction recycles carbon trapped in carbonates by chemical weathering and organisms in the ocean. Subduction draws the carbonates down to the lower crust where they are "cooked" and the carbon converted to $\mathrm{CO}_2$ and $\mathrm{CH}_3$ which are returned to the biosphere via vulcanism. Without this recycling, all the carbon in the biosphere would end up as limestone on the ocean floor. Starved of carbon, all life would perish. 
It appears that the Earth has a much thinner and much more heterogeneous crust than either Mars or Venus. Likely, this is a result of the collision that produced the moon. The thinner crust is easier for geothermal convection to move. Also, it's possible that the collision "stirred" the distribution of heat producing radioisotopes leaving more near to the top of the mantel.  On Venus and Mars, it is likely that the very heavy radio isotopes sank to the core. This would make the upper mantel on Earth hotter than the other planet and hotter than it would have been without the collision. That would drive convection faster. 


Likely, the critical difference for Earth was the collision that formed the moon. That both "stirred the pot" and added more mass. Earth retains both heat and hydrogen. Retaining hydrogen means retaining water which means life and hydrated minerals. Life produces oxygen that protects the water. Hydrated minerals provide the elasticity needed for tectonic motion. Tectonic motion recycles carbon needed for life. 

Take away anyone of these factors and a planets tectonic motion will eventually wind down in less than a billion years.