Well, as a matter of fact, such feelings are very simple and easy. From what I know, your problem is probably resulted from lack of sleep, bad use of your eyes, and perhaps from catching a cold. So you should avoid making those occasions happen, and if such feelings still occur, you must consult a doctor and take actions against your problem, such as taking exercise, getting a better diet, etc.
Reading on computer is not a good idea. You just got tired eyes. Eyes are sensitive human organ. Spend just a little time to relax the strain will be very helpful when you are reading for a long time. It is said that, staring at the computer continuously for more than an hour causes as much strain as about a few kilograms of weight placed on our eyelids. Don't wait to take care of your eyes until them give you warns. In fact, much of the damage would have already been done by that time. There are some symptoms for tired eyes, which should take care in time. Such as what you mentioned: heavy eyes, along with irritating, burning, and red and sore of your eyes.
So, don't put too many strains on your eyes. While reading on computer, remember to take rests for you eyes. You can do some eyes excises, wash you face with cold water etc.
Heavy eyes or tired eyes are one of the biggest prices we pay for lack of sleep. Although it is not a disease, and does not require medical treatment, it makes us look older than others. There are so many reasons and symptoms for heavy eyes. First, working on the computer for long time,for this case, you should take a rest in regular time or do some eye exercises. Second, the light in your room is gloomy, this will bad for your eyes. You should you have the room lit evenly, so your eyes don't have to work as hard. Third, lack of sleeping to cause the tired eyes. It is very important to keep eight hours a day to improve our eyesight. Last one point, you will feel heavy eyes if your glasses or contacts are not up-to-date, for this, you should update your glasses or contacts usually. Don't worry much about your heavy eyes, it will be changed to be better in a short time.