“Can teething upset a baby's stomach?”

Teething can give you symptoms like referred ear ache (due to close nerve sensitivity with the ear), increase stomach acidity due to stress and irritability, this can also give acidic loose bowel movements, which in turn can give you skin irritation in the diaper area. Some people associate teething with fever, in reality there should not be real fever ( >100.4 F rectally or infrared thermometer) with teething. The temperature may raise a little above normal but not fever.
Teething, because of gum discomfort may give infants and children irritability, restlessness and decrease appetite.
The good news is that these issues tend to resolve once teeth erupt and are short lived.
When in doubt call your Pediatrician to consult for the possibility of any other underlying illness.