Dead woody roots take years to break down, and will do so over time. There will be extra fungal activity while that's going on, but none of it is anything to worry about EXCEPT for honey fungus, which does sometimes start growing in association with dead wood, and that really isn't something you want in the garden, because it spreads to healthy, woody plants and will kill them.

If you are gardening in a warmer climate, like mine, the rapidly decaying wood can cause the soil to go anaerobic, killing the roots of any trees or shrubs planted over it. Normally, in warmer climates, it is better to remove the old roots at least 2 ft. deep, and 4 ft. wide where a new tree will be planted. Another problem with lawns and pavers is the possibility of slime molds growing on the dead wood forming spongy mats of sporing bodies on the surface in the summer.