Is it possible to remove red veins in eyes?

This problem can be corrected by a procedure we offer called EyeBright in which we remove the swollen conjunctiva and encourage a new pristine clear conjunctiva to heal in thus improving your appearance. Check it out at Laserfox.com

First you should see an eyecare professional to be sure that the vein isn't a sign of pathology like cancer. Most red eyes are due to common conditions like dryness, allergies and too much screen time. Many people do find relief with LUMIFY which is the first over the counter redness relieving eyedrop that safely reduces redness while allowing oxygen and nutrition to get to the eye, reducing "rebound redness" and other unwanted side effects. One drop of LUMIFY works in about a minute and can last up to 8 hours. Be sure to wait a few minutes before inserting contact lenses if you wear them. 

As an oculoplastics surgeon, I feel confident you first need to make certain there is nothing more than dry eyes making your eyes red.  Treating root causes tends to result in optimal results.  
That said, brimonidine (aka Lumify, a new, over-the-counter product for red eyes) is safe and whitens most eyes in a matter of 5 minutes and lasts 8-12 hours.  It can be used 2-3 times daily.  Also, there are surgical treatments we can do for chronically dilated vessels on the eye....but (again) we do not want to camouflage any signs of an underlying medical disease or illness, so please seek out attention of an ophthalmologist for workup before having any procedure done that puts your vision at risk.