  Fast radio bursts (FRBs) are millisecond bursts of radio radiation whose
progenitors so far remain mysterious. Nevertheless, the timescales and
energetics of the events have lead to many theories associating FRBs with young
neutron stars. Motivated by this, I explore the interaction of FRBs with young
supernova remnants (SNRs), and I discuss the potential observational
consequences and constraints of such a scenario. As the SN ejecta plows into
the interstellar medium (ISM), a reverse shock is generated that passes back
through the material and ionizes it. This leads to a dispersion measure (DM)
associated with the SNR as well as a time derivative for DM. Times when DM is
high are generally overshadowed by free-free absorption, which, depending on
the mass of the ejecta and the density of the ISM, may be probed at frequencies
of $400\,{\rm MHz}$ to $1.4\,{\rm GHz}$ on timescales of $\sim100-500\,{\rm
yrs}$ after the SN. Magnetic fields generated at the reverse shock may be high
enough to explain Faraday rotation that has been measured for one FRB. If FRBs
are powered by the spin energy of a young NS (rather than magnetic energy), the
NS must have a magnetic field $\lesssim10^{11}-10^{12}\,{\rm G}$ to ensure that
it does not spin down too quickly while the SNR is still optically thick at
radio frequencies. In the future, once there are distance measurements to FRBs
and their energetics are better understood, the spin of the NS can also be
constrained.
