Do diamagnetic substances interact with magnets?
It is a very small effect, and they do interact with external magnetic fields. It's usually a weak quantum mechanical interaction, but it can be visible in diamagnetic substances.
When diamagnetic substances interact with a magnetic field, the electrons circulate and form an eddy current. The magnetic field produced by that eddy current is repulsive, opposing the applied magnetic field:
This generates an induced magnetic moment (albeit usually small).
In paramagnetic and ferromagnetic substances, this natural diamagnetism is suppressed by the sheer magnitude of the more usual magnetism wherein we observe more permanent magnetic moments.
And for fun, here's a video of a guy demonstrating diamagnetic substances interacting with a very strong, neodymium magnet (