Yes, sleeping with the fan can lead to dry eyes. As we know that if you do not have the necessary balance of these three layers in your eyes, or if your eyes do not make enough tears to coat the eye, you may have a condition, which is known as dry eye. Also, dry eyes, can be caused or worsened by poor tear quality, poor tear production, not blinking enough and medications. For example, when you use something to treat high blood pressure, allergies, depression and acne, it can have the side effects. But you also should know that a dry climate, dust, smoke, heat, especially sleeping with a fan may make your situation become worse. In some serious cases, it can lead to a humidifier to alleviate irritation. So you should try some ways to treat it. For example, you can use some lubricating drops, but at the same time, you also need stay away from those which can treat red eye, because it will only lead to dryness.
Actually the sleeping with a fan will not cause the dry eyes if you don't put the fan in the direct near position to your eyes. However, the use of fan will increase the movement of the invisible bacterium in the air which may cause the eyes easily to get infection. You should care about this.