The mechanism
Some tattoo pigments — especially older ones and those containing iron oxide — are slightly magnetic or conductive. In the strong, switching fields of an MRI they can, uncommonly, produce a transient heating or pulling sensation, or a faint skin irritation, usually mild and self-limiting.
The evidence
Large reviews put the rate of any MRI-related tattoo symptom well under 1%, mostly minor and brief. Documented burns are exceedingly rare and associated with large, dense body tattoos far more than a sheer lip blush. Guidance from radiology bodies is to proceed with clinically necessary scans.
Practical advice
Clients should mention cosmetic tattooing when booking and to the radiographer, who may apply a cold compress or simply monitor for the affected area. Elective scans could in theory wait until initial healing is complete, but an urgent scan should never be delayed for a lip blush.