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.