Your engine performance requires that all components in the car operate at optimum capacity. If your exhaust pipe has a hole or a leak, your engine performance will be affected. You may not notice some of them if you have an older vehicle because they do tend to be a little noisier and a little less efficient. However, some signs that your exhaust pipe is having issues cannot be missed.
Your car will make some interesting noises if your exhaust has a hole in it. The hole could be caused by metal corrosion, causing the pipe to break down from oxidation. The noises will be unmistakably loud. Another noise you could hear is backfiring. If the air is mixing with the fuel it will create a back pressure and that will cause the backfiring. This reduces the performance of the engine and possibly even fuel economy for those engines which have a fuel injection.
A damaged exhaust can sometimes cause burnt gasses to pass backward and re-enter the combustion chamber. That process will reduce efficiency between cycles. You might experience a reduction in power from the engine if this occurs. These factors all depend on where the leak is located in the exhaust. The further away from the engine and the catalytic converter the leak is, the less chance there is of it doing considerable damage.