Is it normal for someone who just started using a mouth guard to have sore teeth?

Yes: The mouth guard could be fitting too tightly or improperly which could be putting pressure on your teeth. This could cause soreness. It is also possible that it is changing your occlusion which could cause soreness.

Common, Not Normal: The goal of a nightguard is to reduce the pressure and soreness of your teeth, not to increase it. So, if your teeth are sore, that generally means that your nightguard needs to be adjusted a little bit. Do not do this yourself! have your dentist check it and make some minor modifications so that it's comfortable and doing its job. Rarely, the nightguard design has to be altered a bit too.

Home made: Was this home made or made by a dentist? If this is a home made mouth guard i would refer you to your local dentist that has been trained by dr. Dawson or pankey. He can make you a proper mouth guard. If a dentist did fabricate your mouth guard then he may need to adjust it.

Yes: If it's an OTC product toss it out and call your Dentist before you do harm. If it"s a custom appliance made by your Dentist know that it will require numerous adjustments to arrive at a neutral position. If you're sore, call your Dentist now.