You'd be amazed at how often having good chest position comes into play in gymnastics. Most of the time, it's essential in maintaining balance and in proper position to lead into another trick. Here, we will look at a few of these, but these examples should give you an idea of when and where it is important.

We'll start at vault. Numerous times in past features, I've emphasized the importance of keeping your chest up when you hit the board for a handspring-type vault. (Vault: The Punch) I won't repeat too much of what's there, but I'll emphasize that if you drop your chest when you hit the board, you'll smack your face into the vault.
Where your chest is in respect with the rest of your body is extremely important on bars. Most of the tricks rely on good form in order to work. For example, casting out of a kip requires you to be in a pike position with your chest leaning over one side of the bar - this will give you the leverage to get enough power to cast. In a free-hip circle, in order to get the power to circle the bar, once you bring your body back towards the bar, you need to drive your chest backwards and down. This will give you the momentum to shoot you up to handstand. Then there's a layout flyaway dismount (illustration): don't think your upper body doesn't help here. While you keep your body in a layout position, you can lead with your chest to give you an extra "push" in the rotation.
Keeping your chest aligned with the beam is imperative if you want to keep your balance. Everything that you do must be centered on a geometric plane perpendicular with the balance beam (illustration). Finishing a trick also requires you to lift your chest. Take a back walkover for example. If your chest is still down by the time both your feet have landed on the beam, your body is as loose as a wet noodle and you'll easily lose your balance. To avoid this, lift your upper body so that you are solid.
And finally, on floor. Body position is always emphasized on the floor exercise, particularly in the dance elements. But chest position is sometimes essential to tumbling. For example, lifting your upper body as you land an aerial cartwheel (illustration) not only makes it look nicer, but it also makes the trick more solid and sharp. In back tumbling, keeping your chest up (so that your body is in somewhat of a hollow position) will help speed up your tumbling and give you the "set" that is needed for backs, layouts, etc. In forward tumbling, keeping your upper body slightly arched will give you the snap you need to do front handspring fronts/bounders/layout fronts/etc.

So as you can see, your upper body plays a major role in a large portion of how well you perform. Understand the optimal positions to help in your skills - watch others, analyze, and pay attention to what your body is doing. When you become aware of what you need to do, you should soon see some improvements. Good luck!