Is it possible to overcook beans?
In short, yes, beans can be overcooked.
As far as legumes and "real" beans go--absolutely. Beans can still be burnt (such as over-fried falafel, or grilled haricots verts), or overcooked until they just turn to mush. If you simmer bean soup long enough, the beans will just disintegrate and lose their shape.
Dried beans in particular are fairly forgiving--there's a pretty wide range where they are edible & enjoyable. The goal is to get them so they aren't crunchy yet aren't so mushy that they're falling apart.
Fresh beans (haricots verts, green chick peas, fava beans, etc) are quite the opposite. Like other fresh vegetables, the cook time is relatively short and thus it is easy to turn them into a mash by overcooking for just a few minutes. Some people do like mushy vegetables--but many would consider mushy fresh vegetables (including fresh beans) to be a culinary sin.
Coffee beans aren't really beans--they're seeds from a fruit--but they are very temperature sensitive when roasted. There is a wide range of roasts, but dark roast in particular is essentially brought right to the edge and stopped before burning. For folks who like lighter or medium roasts, even dark roasts of coffee can be unpalatably burnt tasting.