There have two reason may cause your red eyes after tanning. One is allergy the other is sun light burning of the cornea. The strong rays of the sun will also lead the injury about the eyes vision and the eye socket of the skin. About how to treat it, the cold compress of clean water and towel could be help to your eyes. Relax your eyes and enough sleep will let the blood circulation goes will with the eyes skin. The next time when you go out for sun bathe, please wearing a pair of sunglasses to protect your eyes.
Well, it is normal to have red eyes because of tanning. And of course, enjoying tanning bed can lead to some eye problems. As you can see, you may suffer red eyes. For your situation, you can use some cold compresses to treat it. Some cool tea bags and ice can be useful. Also, eye drops can be effective too. By the way, because of the uv rays omitted by tanning bed, it will affect eyes, leading to photo conjunctivitis, inflammation of the cornea. So just be careful.
It is easy to get red eyes after tanning if you don't have the protected instruments. You could use the warm compress at home to make the eyes feel comfortable and release the redness of the eyes. You could also use some eye drops with anti-inflammation to make the redness of the eyes release to some degree.