You can find patio furniture made out of all sorts of materials and depends mostly on your budget and how much time you want to spend with upkeep. The reason you don't see most people putting patio furniture directly on the grass is what others have already mentioned. The grass will die/burn out from being in the shade and you have to move them to cut the grass. Furniture in the elements will develop mold/mildew if you don't wipe them off after it rains or even from the humidity in the air. It will also be awkward sitting on the ground because it it isn't a flat surface, the chair will be to one side or might even fall over when someone sits on them. A patio is a better choice for permanent furniture. If you just want to put chairs on the lawn when having guests is different from something that will be set out permanently. Even putting down some crushed stone/lava rock is probably a better option then putting the furniture directly on the grass.
Wood furniture needs periodic power-washing/staining. The sunlight will cause all materials to fade/weather over time. Those new synthetic woods are supposedly more weather resistant but also tend to be pricey. Adirondack chairs are sold in several materials from plastic to wood. All require some kind of maintenance and the cheaper plastic ones tend to fade quickly in the sunlight.
Wood furniture needs periodic power-washing/staining. The sunlight will cause all materials to fade/weather over time. Those new synthetic woods are supposedly more weather resistant but also tend to be pricey. Adirondack chairs are sold in several materials from plastic to wood. All require some kind of maintenance and the cheaper plastic ones tend to fade quickly in the sunlight.