This claim can be subjective based upon what method of self-tan used. Many kinds of self-tan are FDA approved, yet some are not, and can lead to cancer.
The main chemical being used in self-tan lotions is DHA (dihydroxyacetone) which bonds amino acids on the top layer of your skin to melanonids. Many find this chemical reaction to be alarming, but Dr. Beth Goldstein, a board certified dermatologist has found any safety concerns involving DHA, particularly people with altered skin barriers. However she also claimed that given there is little to no penetration beyond the top layer of skin, which is composed of dead skin cels, its likely to not cause any serious issues to those with normal skin.
However, the opposite can also be argued, that self-tan can be the cause of serious health concerns. The National Library of Medicine has found that tanning beds especially can be concerning. Tanning beds use ultraviolet rays in order to add color to the skin, which can just be damaging as the sun, which radiates ultraviolet light as well. One statistic that really stood out to me when they were surveying melanoma cases, where a total of 601 cases were identified. They found that females are three times more likely to have used a tanning bed, and amount the tanning bed users, 90% were aware of the dangers of a tanning bed. This was really interesting to me because ninety percent of those women knew what they were getting themselves into, and how it was almost guaranteed to rise skin and health concerns. (Tanning beds: A call to action for further educational and legislative efforts).
The claim "Is Self-Tanner Safe, and Should You Be Wearing It Overnight?" is simply objective and can't be completely true or false, because self-tan is too broad of a term for a variety if products. However if those who are engaged in self-tan are reading this, please research and consider safer options before hand!