Magnetic field insulators

Mu-metals are your best bet, though superconductors are useful in some situations.

You can think of mu-metals as redirecting the magnetic field, rather than neutralizing it. So if you just have a flat piece mu-metal between the magnet and your piece of iron, you'll still get a reasonable amount of magnetic field -- it just had to "go farther" to get to the iron, so it will be weaker. The usual design for shielding is a box enclosing whatever you want to shield.

Bringing a superconductor close to a magnet, on the other hand, will simply cancel out changes to the magnetic field inside the superconductor itself. This is typically less useful for your problem, but it might be applicable for other applications you may need. In particular, if you need fields eliminated from a cavity inside a material, this could function better than the mu-metal.