Is it normal for your teeth to hurt when waking up with a mouthguard on?

Not unusual: It depends on whether or not the mouthguard fits well and what its' purpose might be. The tooth pain may be from orthodontic pressure on the teeth. This may be intended or may be due to a poor fit. Is the mouthguard intended to treat tooth grinding or clenching, or is it a sleep apnea appliance? More info is needed to give a better answer.

Maybe: If you have a soft nightguard they generally should not hurt. But if you have a hard nightguard it may too tight on a tooth or teeth and should be adjusted.