Is it permissible to suntan on Shabbos?
- The possible prohibition would be tzoveah -- dyeing.
- Is it considered permanent enough to be Biblically prohibited?
- Is it dyeing at all being that no coloring agent is used?
I have seen the Az Nidabru misquoted as permitting. What he actually says is that people who sit out in the sun are only doing it because it's enjoyable and not to get a tan, and therefore it's permitted. Additionally, he invokes "Ain derech tzviah bikach" that this is an abnormal way of dyeing, which is plainly false, it is a normal way of dyeing. He does not seem to be aware of common practice to lie in the sun in order to tan. The clear implication is they if one would intend to tan it would be prohibited.
As far as the Shemiras Shabbos Kihilchasah, he permits on the grounds that since people tan and then become pale based on how much time they spend outside, it is not called dying at all, similar to how opening and closing a door does not constitute building.
If anyone else has more sources to add, I would appreciate it.