Well, I have to say that it is kind of normal to have eye infection after you have used eye drops which are for dry eyes. And there are some evidences which have shown that eye drops can have side effects in your eyes. For example, by using eye drops, they can affect your eyes, leading to redness, irritation, discomfort, burning, stinging or watering of the eyes. For your situation, you may need to go and see the eye doctor as soon as possible. And he can give some advices. Besides, it may also cause the eyelashes to grow and cause the iris to change to a darker color. Anyway, just be careful about your eyes.
Yes, the possibility of getting eye infection by using eye drops to release dry eyes still exists. Your way of using eye drops may not be correct. Or your eye drops may be expired for some time which you could have a check. You may be allergic at this kind of the eyes drops. Thus, you'd better stop using it and have a good rest for the eyes until you get the recovery.