Having had to design surgical instruments to operate in in MRI imaging field (not just the magnetic field, but in the patient during a scan), I can answer this.
The warning on the machine is overly cautious. Almost any material can be put in in MRI, but should be tested. 99% would probably fail (and 99% of those can be predicted to fail). And who wants to test every friggin' thing someone wants to bring into an MRI room. Best just to say "NONE!"
But for stuff designed for the MRI, it's a different story.
There are 2 levels of compatibility: MRI safe and MRI compatible.
MRI safe means it won't hurt anyone. Don't use large pieces of ferro or paramagnetic materials, or the magnet will apply significant force. In other words, just don't use steel (plus a few other exotic alloys). And don't use long, thing wires, or you can create some induction heating from the radio waves (the R stands for resonance
MRI compatible means that the material won't disrupt the imaging field. Either by warping the magnetic field, messing with the radio waves, or creating discontinuities (the FFT algorithms create artifacts from abrupt hydrogen density changes. Air/interface is enough sometimes).
Turns out polymers are ideal, followed by ceramics. But a small group of diamagnetic metals work pretty well too (but still have dimension and geometry constraints). Gold, silver, lead, and copper work great. BeCu alloys can be used and replace steel.
A small item (20 mm x 5 mm dia) made of copper should be MRI safe. MRI compatible is something that has to be tested, but copper/polymer objects might not screw with the image too badly if designed well. And it makes great sense to tap the radio field for a little bit of power.