  We study the sensitivity of the $z=0.1$ Lyman-$\alpha$ Forest observables,
such as the column density distribution function (CDD), flux PDF, flux power
spectrum, and line width distribution, to sub-grid models of active galactic
nuclei (AGN) feedback using the Illustris and IllustrisTNG (TNG) cosmological
simulations. The two simulations share an identical Ultraviolet Background
(UVB) prescription and similar cosmological parameters, but TNG features an
entirely reworked AGN feedback model. Due to changes in the AGN radio mode
model, the original Illustris simulations have a factor of 2-3 fewer
Lyman-$\alpha$ absorbers than TNG at column densities $N_{\rm HI}< 10^{15.5}$
cm$^{-2}$. We compare the simulated forest statistics to UV data from the
Cosmic Origins Spectrograph (COS) and find that neither simulation can
reproduce the slope of the absorber distribution. Both Illustris and TNG also
produce significantly smaller line width distributions than observed in the COS
data. We show that TNG is in much better agreement with the observed $z=0.1$
flux power spectrum than Illustris. We explore which statistics can disentangle
the effects of AGN feedback from alternative UVB models by rescaling the UVB of
Illustris to produce a CDD match to TNG. While this UVB rescaling is degenerate
with the effect of AGN feedback on the CDD, the amplitude and shape of the flux
PDF and 1D flux power spectrum change in a way distinct from a scaling of the
UVB. Our study suggests that the $z=0.1$ Lyman-$\alpha$ forest observables can
be used as a diagnostic of AGN feedback models.
