Dilated veins are there to stay, short of surgery. Sometimes, dilated veins can be a sign of inflammation or allergies. These can be treated. But sometimes, you may just have large dilated veins; and they are permanent. Visine only temporarily constrict them so that the eye appears more white.
If you have a clump of dilated veins, they may be a hemangioma and can be surgically removed.
Hope this helps and good luck.
They are not dangerous to the health of your eyes, only to the vanity of your appearance.