Do Viruses produce a biomagnetic field?
Every atom produces a magnetic field, so the formally correct answer would be "yes" (assuming that viruses belong to the tree of life, which is disputed -- otherwise, one would not use the prefix 'bio').

However, biomagnetism as a science (and not pseudoscience) is concerned with more measurable effects. The biomagnetism of a virus will be negligible compared to all the surrounding cells, and hardly measurable.