There are many chances and situation that people get something else into eyes. I once get hot spicy oil, makeup remover and sweat into eyes. Sweat is the least irritant thing in all of them. Most part of sweat is water. At the same time it contains some minerals, for example sodium, vitamin C, uric acid and etc. When we got sweat into eyes, we might feel a sharp ache and we couldn't help blinking. Blinking is the instinct reaction to the sweat in our eyes. Because our gland which could produce tears is in the upper eye lids, blinking could help tears to get to the surface of our eyeballs, washing the eyeballs and kill the bacteria. So I don't think it is a big deal to get sweat into eyes. No need to worry too much.
Well, generally speaking, a drop of sweat will not cause eye infection in your eyes. And of course, if you have got too much sweat in your eyes, it can be harmful to your eyes. So just be careful about it. Besides, you should know that there are also other reasons which can be harmful to your eyes, and even lead to eye infection. For example, makeup on the surface of the eye or a reaction to solutions can vary from slight to marked redness of the eye with slight to marked pain. And eye infections can cause marked redness, light sensitivity, blurred vision, and a variable degree of pain. So it will be dangerous.
Well, it seems that you are quite concerned about your eye health. So, as far as I know, if too much sweat get into your eyes, you are likely to have eye infections, because there is some bacteria and dirt in the sweat. Of course it would be harmful, leading to some uncomfortable feelings or painful eyes. What you need to do is to get some eye drops . But your eyes have got some immune ability, so take it easy.