Does a bad sinus infection cause eye pressure to rise?

No: No, it can feel like a "pressure sensation" behind or in your eye, but the actual pressure inside the eye is not affected by a sinus infection.

Not usually: While the pressure in the sinuses can build and cause pain, this isn't transmitted to the eyes as they are separated by bone. Even when the bone rarely erodes and an orbital infection occurs, the pressure usually won't change until it becomes severe.