My little boy was never really sick whilst he was teething, so it must just be because he is putting his hands in his mouth. Either that or his milk is upsetting his tummy.
My little boy got sick while teething
So far, my son has been sick the week before a tooth has come through. I was told it's quite common by the GP. Due to lowered immune system and, as you've said, putting hands and everything else in their mouth.
My son always got sick while teething.