Yes, your adrenal fatigue will cause the dry eyes because of the weak immune system and infection at the body. Your eyes will not only be dry but also be red. You should better have the good rest for the eyes, not sitting in front of the computers for a long time. You'd better use the warm compress and eye drops to moisture the eyes.
Actually, dry eyes are caused by several factors. Your living habit and living environment both matters when considering the causes of dry eyes. For example, if you are used to using computers for a long time and sleep very late, you are likely to have dry eyes and red eyes. Besides, dry weather may worsen your condition. Adrenal fatigue often happened when you are under stress and pressure. However, there is no direct connection between adrenal fatigue and dry eyes.
It is possible that adrenal fatigue can cause dry eyes. It is possible, but the chance is very minor.
Adrenal fatigue may cause dehydration which result in dry skin, a lack of water inside the body and result in Sjogren Syndrome which is featured with dry eyes.