No... Well, yes. I say no because you are in a machine that kills germs. UV light is used in all types of situations, including hospitals, to kill viruses and bacteria. HIV only transmits through blood, semen, vaginal secretions. You can not catch it from sweat, which is the most likely bodily fluid you'd come in contact with in a tanning bed. Even if there is a small amount of HIV virus in some fluid in the bed, unless you have some sort of open sore it can not get into your system.
Just by proxy of the tanning bed being what it is, it is going to kill anything that shouldn't be there. That includes all types of viruses from streptococcus, influenza, HIV, herpes, Chlamydia, pubic lice, scabies, to rhino virus. The bed it self my be dirty, which is why you can just ask for it to be wiped down in front of you.
HIV and other STIs are not something you are just going to catch from a toilet seat, or tanning bed. Shaking hands, kissing (except herpes), urine, swimming, playing cards, nor most any innocuous activity. This is not something anyone need to worry about at all.
If you are consumed by the fear of catching something from a tanning bed or a toilet seat, just wipe it yourself. Carry Clorox wipes.