Can you "overdo it" when working out? Of course you can, and at some point we all do to a certain extent. You want to make sure that plain old soreness, not pain, is all you get. There is a reason people ease into these workouts. Because of the fact that CrossFit produces a strong adaptation from your body, there is a potential to get a negative systemic response if you go too hard when starting out.
A condition called rhabdomyolysis can occur with extremely hard efforts or workouts. This is a body wide condition that starts in the cells of your muscles while you work out. When the muscle cells get overloaded with oxygen debt, they start relying on other means of energy production. When that energy runs out, the muscle cells enter into a dysfunctional state and some start dying (like you're probably feeling during that workout).
Potassium and calcium inside the cell shift to levels that cause certain functions inside the cell to completely shut down. All scientific terms aside, that's when it truly sucks to be you. Your body can handle a certain amount of muscle damage from workouts, that's how we "grow" our muscles, by taxing them. Once you pass a certain point with this muscle tissue loss there is a potential for very significant problems.
CrossFit workouts demand power from your muscles, (that's force times distance over time; for more power - a shorter time). If you overdo it you can cause so much muscle damage that the waste materials end up in your kidneys, where normally those wastes go, but in much smaller amounts. If you have trashed your muscles tissue by exerting too hard, you can have serious kidney problems. In some rare cases this has led to death. You can avoid this by using common sense and following one of the most basic rules of CrossFit: Leave your ego at the door.
How do you avoid Rhabdo? Stay hydrated (not 20 minutes before a workout, more like 12 hours), drink alot of water every day whether working out or not. Practice good nutrition habits. Don't starve yourself or eat trash. You need natural minerals and vitamins. They don't all come from a designer shake and you need the fiber from a normal vegetable intake as well. And chiefly, ease into this. CrossFit will be here for the rest of your life, enjoy it. There will be plenty of time to get your butt kicked by fun/hard workouts!