Baby oil (affiliate links used in this post) and iodine for tanning? Was (is) that a real thing? And was (is) it safe? The answers are yes, it was (is) a real way of tanning, and no, very much no, it was (is) not safe.
Modern sunscreens didn't begin appearing until the 1970s, and those with very high SPF ratings came even later. Before that, beginning in the 1950s, there were commercial "sun tan lotions" that had virtually no protective qualities.
And before that, some people mixed iodine into baby oil and slathered that on themselves. Some people still do it today. The reason for the baby oil is clear: it concentrates the sun's rays and leads to a faster, darker tan. (That is also why you should never use it: fast, dark tans are dangerous.)
Why people added iodine to the baby oil is harder to pin down. Some sources claim the iodine deepened the penetration of the sun's UV rays into the skin (again: that is a terrible thing you do not want to do!). But what the iodine was probably doing was coloring the skin a bit — adding a bit of a spray-tan, so to speak, on top of the real tan that came from using the baby oil.
Using the iodine is not just a terrible idea from the perspective of skin damage and possible future skin cancer, however. Iodine is a drug, and like all drugs it comes with possible side effects. The more of it you put on your skin, the greater the possibility of having a side effect, and the greater the possibility of that side effect being bad.
So, yes, a baby oil-and-iodine combination really was used for tanning (and some people still do use it). But doing so was and is a terrible, terrible idea. Do not do it!
A tan is nothing more than skin damage, and any amount of tanning is too much. If you must be in the sun for extended periods, protect your skin, don't fry it.