MRI is one of the safest medical imaging procedures available.
Unlike X-ray imaging or CT-scanners (which use X-rays), MRI’s do not use ionizing radiation, a type of radiation which causes genetic damage and leads to increased risks of cancer.
MRI imaging, by contrast, uses magnetism and radio frequency (RF) energy, plus some very clever advanced physics, to generate images. To date there have been no quality studies that demonstrate that either magnetic fields or properly administered RF energy have any dangerous long term effects on our biological bodies. This is not to say that MRI imaging is perfectly safe and ‘foolproof.’
The magnets needed for MRI imaging are typically tens of thousands of times more powerful than the Earth’s own magnetic field. These uber-powerful magnetic fields can act on magnetic materials nearby (as in things you carry with you, like a nail clipper or tool) and turn those objects into magnet-homing missiles. The same magnetic forces also penetrate your body and can act equally powerfully on magnetic objects within you (such as aneurysm clips, or pacemakers, or orthopedic implants).
For these reasons it is critically important that every person and object entering the MRI area be comprehensively screened for magnetic / metal materials before they’re allowed to approach the MRI’s giant magnet.
There are other safety concerns attributable to the administration of the MRI exam, itself… things such as the proper settings for the scan, positioning the patient correctly so that they don’t rub up against the scanner, and making sure that the patient has appropriate hearing protection (these scans can be extraordinarily loud).