Can teething cause coughing in a 3 month old baby? when must you take the baby in to get checked? he has no fever?

No : I don't expect teething to cause much of any symptoms beyond more mouthing of fingers/chewing. Cough at that age is generally a reaction to irritants they breathe from their environment. Dust/fragrance/fumes/smells/etc. A nasal cleanout with saline nose drops helps most. If feeding falls off & normal activity diminish it mayy be worthwhile to have the kid seen.

No association: There is no association that I know of between teething and coughing. Cause of coughing probably environmental. Call Pediatrician's office, report symptoms, ask if they want to see child. Child should be on a carefully regimented well-baby/inoculation schedule, and may be overdue. Should also see specialist Pedodontist once 1st baby tooth erupts, but no later than 1st b'day.