No scientific evidence shows having tattoos increases the risk of skin cancer or recurrence. However, if the tattoo is too close to or within the nevus, it may have a higher chance of evolving into a melanoma or another skin cancer.
I have tried tattoos with red and black ink on mice in combination with UV until the mice developed skin tumors. Surprisingly, the black tattoos had a protective effect against skin tumors - it was like a permanent sun screen, which actually make good sense since it aborbs the UV radiation. The red ink did not have the protective effect and a little tendency that the tumors developed faster. We have also measured degradation products from the tattoos and there were some in e.g. lymph and liver cells. However, nothing to support higher risk of skin cancer...