That sweat gets into eyes will cause redness of eyes, yet usually it won't cause eyes infection. Sweat is not very irritant to our eyes. It consists of water and some other minerals, including sodium, uric acid, and even vitamin C. It is not harmful to your eyes as you imagined. Eyes are really delicate and soft. When your sweat gets into eyes, they will protect them instinctively. That is why you blink your eyes unconsciously. The gland which could produce tears is in the upper eye lid. Blinking helps transfer the tears to the surface of eyeball. Tears are very precious. They could help clean and wash our eye ball. They are very important lubricant. Without them, we would feel very gritty and uncomfortable whateven movement we make. They could kill 90-95% of the bacteria in our eyes. So just a blinking would help you solve the problem you met. Yet if you still worry about it, you could use some running water to wash your eyes too. The redness is just temporary. It will last for only a few minutes. Don't worry too much.
Well, it seems that your eyes are not in good conditions now. So, as you can see, slight amount of sweat is just ok, but excessive sweat could give rise to some discomfort and maybe painful feelings,and if you rub, you get red eyes.. Of course it is temporary because your tears would wash your eyes automatically and fix the problem. However, if the situation is quite bad, you need some antibiotic eye drops to deal with the problem.