From the Guidelines

No, an MRI scan is not safe with a foreign object, such as a bullet, in the body.

Foreign bodies in tissue may become dislodged, leading to damage or haemorrhage, which is a particular hazard in the eye or near blood vessels 1.
Ferromagnetic objects within the 3 mT field contour will experience an attractive force, pulling them towards the centre of the magnet and a torque as they attempt to line up with the field, making them potential projectiles 1.
Even small objects can become dangerous projectiles, sufficient to injure or kill anyone in their path, and larger objects can trap or crush a patient or staff member 1.

Metallic implants such as hip prostheses, mechanical heart valves, coronary stents, and sternal sutures present no hazard since the materials used are not ferromagnetic, although a local image artifact will result 1.
However, care is required in patients with cerebrovascular clips as certain older models remain unsafe in the MR environment 1.
Electronic implants, such as pacemakers and implanted cardioverter-defibrillators, are considered a contraindication, although recently there have been several case series of patients with pacemakers who successfully underwent MR scanning at 1.5 T with very few changes in pacer function 1.
It is essential to carefully screen equipment, staff, and patients before entry to the magnet room to prevent any potential hazards 1.

From the Research

The safety of Magnetic Resonance Imaging (MRI) scans with foreign objects, such as bullets, in the body is a concern due to potential interactions with the magnetic field and radiofrequency pulses 2, 3, 4, 5, 6.
Studies have shown that bullets and shotgun pellets made of lead with copper or alloy jackets appear to be safe with respect to MRI-related movement at 1.5-, 3-, and 7-T static magnetic fields 3.
However, bullets and pellets containing steel may pose a danger if near critical body structures due to strong magnetic field interactions, which can cause movement, heating, and artifacts 3, 5.
Ferromagnetic bullets can readily rotate within a gelatin phantom in response to magnetic torque and produce severe MR artifacts and image distortion 5.
Non-ferromagnetic bullets and pellets demonstrate only mild to moderate metal artifact during spin-echo and gradient-echo magnetic resonance (MR) imaging 5.
Temperature increases of selected ballistics during 3-T MRI were not clinically significant, even for the ferromagnetic projectiles 3.
It is essential to exercise caution when performing MRI scans on patients with retained bullets or other metallic foreign bodies, particularly if they are located near vital neural, vascular, or soft-tissue structures 3, 5.

MRI technologists require sound knowledge of the physical principles of the MRI scanner and must understand the associated safety risks and how to avoid adverse events from occurring 6.
Rapid advancements in MRI technology and other correlated areas, such as medical implant technology, place increasing demands on the MRI technologist to constantly keep abreast of current and future developments 6.
Patients with retained bullets or other metallic foreign bodies should be carefully evaluated before undergoing an MRI scan to ensure safe practice within the MRI environment 3, 4, 5, 6.