Can a toothache be caused by being sick?

No: A toothache is only from the tooth, and not due to a sickness or fever.

Sometimes: Very few illnesses can cause a toothache. However, acute sinusitis can cause inflammation that is felt across the roots of the upper molar, and occasionally the premolar teeth. It is not a "toothache" by definition, but sure feels like one if you are suffering from it. A radman test can be done by your dentist to find if the pain you feel in your teeth comes from the teeth/gums or the sinus.

Vice Versa: No, but you can get sick from a bad tooth. See dentist for evaluation/