Can teething cause my child to become ill?

No. Children experience continuous tooth eruption, on average, between six months and thirty-two months of age. Coincidentally, they lose maternal antibody protection and begin to socialize at this same time. So, it's no wonder that signs of illness, and in particular fever, runny nose, diarrhea and rashes spawned the teething myths of yore. But, don't argue with your grandmother about it ...
No. Teething does not cause an illness in the child. However, children can catch colds, flus, strep throat infections, etc. . . , while teething; so both an illness and teething can happen together at the same time.
No. There have been a great misunderstanding about teething, and teething associated pain/discomfort has been way blown up. Teething has been wrongfully blamed for the common maladies in infants of 6 to 12 months age such as fever (febrile illness), diarrhea or even rash. It would be reasonable to expect a mild elevation of temperature not above 101 degree f and some fretfullness.
No. Teething should not cause significant irritability, flu like symptoms or high fever. It may cause increased drool, low grade fever and some increase in fussiness. Rarely your baby may get a gum infection from teething. If the gums look increasingly red and have discharge or odor see your pediatrician. Teething is a natural process and should not cause illness by itself.