like BDD?
If there is an allergic reaction to tattoo pigment, then one must be very careful with laser treatment because it could cause the pigment to spread into the body and cause a generalized allergic reaction. Sometimes tattoos with allergic reactions can be excised if they are small enough. This could possibly be covered by insurance. It would be very unlikely for insurance to cover laser treatment for an ordinary tattoo. I have had insurance cover laser treatment for an embedded fireworks "tattoo" but that is a rare circumstance.
The vast majority of medfcal insurance companies do NOT pay for cosmetic surgery (defined as procedures which do not save life or improve function). It has been proved in multiple studies that Cosmetic Surgery dos not correct certain psychiatric conditions such as Body Dysmorphic Disorder (BDD) but may actually worsen it.