Well, yes, eye makeup can just lead to some eye infection in some situation. Generally speaking, by using eye makeup, it will be good for your eye health. But on the other hand, there are also some people who suffer eye infection from eye makeup, because the eye makeup is made of chemical. And chemical can be the irritants to the eyes, leading to eye infection. And then, red eyes may occur. So when it occurs, you should stop using it. also, go and see the eye doctor.
Yes, it is true that the eye makeup may lead to the eye infection if you don't deal with it properly. You should not put the material of eye shadow or the eyeliners in the eyes which will irritate the eyes and let your eyes feel uncomfortable. You'd better wash the eyes clearly after the eye makeup to avoid the eyes problems. You should keep the eyes clean and moisture.
OK, I regret to tell you that it is true, your assumption that eye makeups could give rise ti infections could be possible. Because as you can see, those makeups are made of chemical stuff and are highly irritative to your eyes. You are likely to get some of those things into your eyes, thus, infections arise. Anyway, try to wear less of those makeups and take care of your eyes. Not only makeups make you look nice and beautiful.