If you do not eat enough protein, this will attack muscle tissue. Also protein is vital in keeping your blood sugars at a normal level, without protein, you may develop some issues there as well, and don't want to have to me on a glucose monitor. Protein also helps carry oxygen to blood cells. You do not have to eat meat or dairy to get all the protein you need, there are a combination of vegetables and legumes you can eat, as well, soy is a wonderful source as are nuts and peanuts
In most instances, you will get muscle wastage, as you body uses muscle already on your body for vital functions. If protein deficiency continues, you would develop kwashiorkor. This is when you have so little protein, that essential blood proteins are low (like albumin) and tissue fluid accumulates in the abdomen. This is why when you see pictures of starving african children, they have very emaciated bodies but swollen bellies.