How to remove a tone/interference from a signal, without using classical linear filtering
A sharp edge or transient may require a very wide-band spectrum to represent it.  Thus any classical or frequency domain filter that completely removes a spectral band could unsharpen or "smear" a sharp transient, by removing a necessary component of that transient.

There are two non-linear methods based on analysis/resynthesis techniques that might work.  Either estimate your tone or interference signal precisely enough to synthesize it and subtract it.  Or characterize your desired signal well enough to resynthesize it without the noise.  You will need a very accurate model for either of these.  Both of these will only remove or add the appropriate fraction of energy in any spectral band, instead off all or none, or some random percentage (relative to the actual S/N) given by a filter response curve.