Will using magnetic base test indicators with breadboards cause electrical interference?
A current can be induced in a wire by a time-varying magnetic field. If the field itself is not time-varying you can also induce a current by moving a wire with respect to the magnetic field.

In your case there appears to be no movement so no current would be induced. Furthermore, if you place the magnets onto a steel base most of the magnetic field will be contained within the gap, if any, between the magnet and the plate so you would need to have wires very close to the magnets to see any effect at all.

The circuit you want to build does not appear to have any components that are themselves influenced by a constant magnetic field, such as Hall effect sensors. (I couldn't see the schematic without registering and I'm not going to do that.)

So, don't worry. Have fun.