If your eyes are not properly protected in the tanning bed or you open your eyes and are exposed to the UV rays in the tanning bed then your eyes can suffer some bad damage and possibly cause blindness.
The eye damage that a tanning bed can cause to exposed eyes or not properly protected eyes in the tanning bed can cause things such as photokeratitis (a sunburn on your cornea), cataracts, skin cancer around the eyes and corneal infections and conjunctivitis as well as macular degeneration.
As long as you wear your eye protection your eyes should be perfectly fine and avoid overuse of the tanning bed which can cause skin cancer and other skin problems.
Tanning beds are safe in moderation however overuse of the tanning beds can cause problems.