Yes, using eye drops may cause ringing in the ears because of the interlinked eyes nerves and ears nerves. You may also find that the eyes and throat parts are interlinked too. When you use the eye drops, you need to use the suitable amount of it. Or else, other parts of the body may get the affection.
I don't think that your ear ringing coming from eye drops. The main components of eye drop are sadium and water, which are very mild and will not cause any allergic reaction normally. For different purpose, it contains different medications. Eye drop is commonly used to control infection or inflammation or allergy with steroids, antihistamine or some other drugs. Some eye drop even contains no medication in it. No matter what kind of medication it has, it is surely less harmful than other oral antibiotics medication. It has some side effects with its benefits, but it is not very serious problem. Usually people might feel a stinging in eyes when instilled the drops. Some people might experience symptoms similar to a cold, such as soar throat. But you should let your doctor know your medical history to ensure no typical and severe allergic reaction triggered such as persisting fever or skin rash or breathing difficulty. If you keep hearing some ringing, I suggest you to visit the doctor to find out what is wrong with your ear.