Well, generally speaking, it is hard to say that the red veins in your eyes can be permanent or not. It all depends on the causes. There are some people who have red veins in their eyes, when they are born, so for them, it can be permanent. But your situation, I think you should not worry about it. it can be removed by some ways. For example, you can have a tea bath. Just have a cool teabag on your eyes for a long time, and then it can be good. Also, you should know that the real problem is probably your eyes being strained. If you watch tv or look at a computer screen a lot your eyes tend to get lazy and strained. so it can lead to red veins. So You can do eye exercises. Or have some eye drops which can also be effective. Anyway, having a good rest for your eyes is essential.
The red veins in eyes can be the result of overuse of whitening drops,an autoimmune condition called Uveitis or an inflammation of the structures of the eye, including the sclera. However, the red veins are not permanent. I would give you some suggestions. You can consult an eye doctor about your condition, since you said you just get tiny red veins, to see whether your condition can get control. You can mention about uveitis and discuss this with him.And you will need some medications. At the same time, you can use warm fomentation for the eyes with heating a dry towel, hold it over the eyes twice a day.
They will not be permanent if you do the protection for the eyes properly. The small tiny red veins appear in the eyes because of the long time use of eyes in front of the computers. You'd better have a good rest for the eyes and use the eye drops to release the symptom. The eyes should use the medicine to anti the inflammation. Thus, the red symptom will be disappeared. If the red veins are not absorbed by the corneal, it will become the yellow points in the eyes in the serious situation. Then you will have to go to the hospital to have a treatment. Keep in mind that having a good rest for the eyes which is the most important thing.