How long does it take for a red dot on my eye to go away?

About 2 weeks. Subconjunctival hemorrhage (a small bleed on the surface of the eye) goes away in about 2 weeks. If the red dot is a blood vessel loop or being caused by something else, it may not go away at all. If the red dot is inside your eye, you should have it evaluated.
About a week. If, by "a red dot" you are referring to a sub conjunctival hemorrhage . However if the redness seems to get worse, is accompanied by pain, discharge or a change in vision, you should be examined by your ophthalmologist.