The fine roots simply contribute organics to the soil and microbes, macro-invertebrates, and even worms will recycle them. The large roots come out with the plants and get composted (often not in the tower but in my compost pile, but a portion of them can go back in the tower). The plants that will create some issues are those with woody root systems, almost all of which are perennial. For a first time gardener, we don’t recommend perennials in the Garden Tower because some of them can create a lot of root mass which can be a challenge to deal with. However, an experienced gardener can use perennials they know they want to come back each year and plan accordingly. The only weedy perennials we’ve had that really become a problem are the woody mints.
It’s important to note that the breakdown of roots often uses a lot of nutrients, so nutrient limitations are more likely to cause growth issues year to year than the actual root accumulation. Some nutrient tea’s can be made with inexpensive organic plant food mixes to supplement the tower should the composting not sufficient in itself — which for some plants, it simply cannot be (heavy feeders like tomatoes that pull a ton of phosphorus out of the soil).