Why do my eyes burn when I'm tired?

Dry eyes. Eyes don't truly get tired, but oftentimes your tiredness can cause or worsen dry eyes. When we get tired, we often stare off or force our eyes to stay open in an attempt to keep from dozing. This can cause the tears on the eye's surface to evaporate as well as decrease the amount of new tears that blinking would normally put onto the eye. The result is the burning sensation of dry eyes.
Fatigue. Might be due to eye strain, dry eyes, lack of sleep, irritation, allergies, contacts, etc. Could be related to blepharitis (clogged oil glands) which is common.
Nerve endings. When your eyes feel tired, this is often a symptom of evaporation of the tear film, or dry eye. The surface of the film breaks down and you get little dry spots. Then the many corneal nerve endings are stimulated because the surface cells are drying out and you feel a burning sensation, which causes you to blink more.