Yes, like any other expensive device, careful engineering is required to minimize unwanted noise in the signals.
For x-ray computed axial tomography (CAT), normal motors are required to rotate the gantry around the patient, and an oil pump with cooling fans on the rotating platform to run a heat-exchange loop on the x-ray tube's anode.
A motor built into the x-ray tube rotates a large anode disc to allow high-power anode dissipation for short times, since typical scans are done while the patient holds his breath; if the tube's anode is out of balance, this will degrade the image.
Modern CAT scanners have a high-voltage supply on the rotating platform for the x-ray tube, and it must be shielded and filtered appropriately.
Most CAT scanners have the detector array (now two-dimensional) rotating on the same platform with the tube, with fancy data communication from the detector electronics to the stationary platform.
As posted above, MRI machines do not require motion (except for the patient couch), but the pulsed gradient currents could interfere with electronic signals; the basic signals from the patient go through a resonant "receive" coil, tuned to a much higher frequency.
Sometimes, motors are required to operate vacuum-variable capacitors to tune the high-power RF coils: these motors are an engineering problem to avoid loss of homogeneity in the DC field due to ferrous components near the main magnet.
Again, as mentioned above, the clearly-audible noise results from interactions between the gradient fields, the DC field, and the coil structure. Also audible is a mechanical refrigerator that often accompanies the magnet to minimize loss of liquid helium.