Something may irritate it and the reaction the eye has is that it burns. Let's say something got in it, then it burns because something is touching the eye that irritates it and it burns.
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It’s better not to. Coffee in general will effect your bloodwork, and it is no different with black coffee. This is due to the caffeine in it which might skew the results. It’s also a natural diuretic, so your bladder will not last as long.
Hope this was helpful!