Can eye infection cause sinus infection?

It is possible. Tear drain into the nasal cavity and an infection in the eye can theoretically spread to the sinuses. More likely the same virus can conjunctivitis, rhinitis and sinusitis.

There are certainly connections between the eyes and the nose, nasopharynx and sinuses. Whether an infection in the eye (bacteria, virus, fungus, parasite) is likely to cause a sinus problem is another story. Whether you're having an eye infection or a sinus infection or both, you should seek medical attention for evaluation and treatment