First of all, you need to check if there are any irritants in the eyeliner that you are using. Low quality eyeliner may contain some cheap and irritant chemical which damages your eye skin.
If you are sure there is nothing wrong with the elements of the eyeliner, you may have been allergic to a certain ingredient in the eyeliner. Test if you are allergic by applying the eyeliner to the inside of your elbow. After a night, see if the skin on your elbow is affected.
Lastly, you are likely to have slight eye infection. In this case, have your eyes examined by the doctor and take some medication.
Well, from what you said, I can see that there are some slight problem with your eyes. As you can see , eyeliners are made of chemical stuff, which might be irritative and harmful to your eyes, leading to some burning eyes, watery eyes, etc. Thus, it would be wise for you to wear less of them and get some anti-biotic eye drops to use. Also, try to get more rest before you are recovered.
Yes, it perhaps if you get a bit of the eyeliners of poor quality into your eyes. As we all know, cosmetics, cream, and eye makeup products contains some harmful metals or other synthetic ingredients irritating our eyes, although they deny it. Besides, the incomplete discharge of the make-up or the remover products do not work as they claimed leading to the dust and chemical residue in or around your eye. As long time accumulated, your eyes hurt absolutely. So good brand products such as eyeliner and special remover are recommended, especially for sensitive eyes.