Hopefully someone can give a better answer than this, but as I understand it, without plate tentonics there is concern that most carbon could get trapped in the crust leaving little in the air for plants to use. However, shield volcanoes could recycle crust without plate tectonics, although a thicker crust may mean a greater amount of carbon could be trapped in it. But even on a geologically dead world carbon might still cycle through ground water, assuming there is liquid water on the planet. On earth life is typically limited by lack of phosphorus or other materials other than carbon. I don't know how much carbon is required for a rich biosphere.