Teething in babies can cause the baby to break out with a rash on his/her face but the rash is usually because of the drool that contacts the babies face when teething and if the drool is wiped off the babies face often enough it shouldn't cause a rash on the babies face.
A babies face can also turn red or look flushed because of inflammation of the gums which is normal as well in a teething baby.
It's also possible for the baby to have hives that could be be triggered by the teething but the hives aren't a direct result of the teething.
Hives are normally caused by allergens, colds, viral infections or bacterial infections.
If your baby is breaking out in hives during teething then you should take your baby to his/her pediatrician because something else is going on that needs treated.