Can antibiotics make you to have tired or sleepy eyes?

To some degree, yes. Some medications can dry the eyes. These include blood pressure pills, antidepressants, and many others. A good addage to follow is that all medications can cause a variety of side effects, even if it is not specifically written on the package inserts. Dry eyes in turn can be experienced as tired or sleepy.
Sometimes. Many antibiotics can dry out mucous membranes including the ocular surface. It is important to stay well hydrated while taking antibiotics. Tired feeling eyes is one symptom of dry eye, and you can use a topical otc tear substitute several times a day while on the antibiotic to see if this gives you relief.