Eye drops should generally be stored in a cool dry place and for some drugs, especially chloramphenicol, the most commonly used ocular antibiotic in Australia, it is preferable to keep the bottle in the fridge.
Patients should not keep their eye drops beyond the printed expiry date. The current policy is that once eye drops have been opened they should be disposed of after 28 days. This is based on research from earlier times when drops were dispensed in glass bottles with glass pipettes, and many eye drops did not contain preservatives. To my knowledge none of this research is current, using modern dropper-type bottles. This policy seems a terrible waste and causes increased expense to the patients and the health system.
Although evidence is needed to support the practice, some ophthalmologists allow patients who are using drops regularly to keep the bottle for up to two months (although most of them run out after about six weeks). Of course when patients have drops that they use only from time to time, such as artificial tears or other drops used purely for comfort, then these drops should not be kept long term as the risk of contamination may then be significant.
Patients who develop an allergy or other reaction to the preservative in eye drops may need to use a formulation without a preservative. Many eye drops are also available in this form in single-use disposable containers.