What are the disadvantages to exercising with sore muscles?

Muscle recovery: Muscles work best if they are challanged enough to break down a bit (catabolism), then given time to recover in keeping with environmental demands (anabolism). That is what muscle building is all about. Some soreness is good, it is a sign of strengthening. But intensifying sharp pains with progressive weakness is an indication that you are overdoing it.

Overtraining: After weight training, your muscles need to recover in order to build. The presumption is that if they're still sore they're not healed, and continuously battering muscles that haven't adequately recovered will lead to overtraining. You'll lose rather than build muscle. Try training muscles once weekly to allow for adequate recovery.