There must be some problems for your eyes although the nose, throat and eye nerves are interlinked. When your eyes get red, you must have some problems, like the inflammation in the eyes. You'd better go to see the doctor and check the eyes. If your eyes get the infection, you should treat it as soon as possible.
Don't worry, it is not red eyes. It just make your eyes appear red for a while. It is normal sign of yawning reflex. It will entry more oxygen and increasing the blood flow to organs when you yawning. And the increasing of the blood flow may cause the eyes red temporarily. But it is not red eyes and will recovery not long time ago.
Well, what you said is really funny. Because everybody yawn every time we get tired and exhausted, and we have no red eyes. So, I feel that there must be something wrong with your eyes which have been mistaken for yawning. Generally speaking, yawning will not lead to red eyes. What I recommend to you is to consult a doctor and find out the possible causes. Just do not jump to conclusions and ignore the seriousness of your symptom.
It can cause red eyes. People yawn to expand respiratory movement and to increase oxygen in the blood and exhaust more carbon dioxide, hence restoring themselves. When one is sleepy, he yawns continually to indicate a rest. So it is normal to have red eyes when you yawn, it is just an indication of tiredness and you need to take a rest.