  We consider interference effects in the production via gluon fusion in LHC
collisions at 13 TeV and decays into $\gamma \gamma$ and $t {\bar t}$ final
states of one or two putative new resonant states $\Phi$, assumed here to be
scalar and/or pseudo scalar particles. Although our approach is general, we use
for our numerical analysis the example of the putative $750$ GeV state for
which a slight excess was observed in the initial LHC $13$ TeV data. We revisit
previous calculations of the interferences between the heavy-fermion
loop-induced $gg \to \Phi \to \gamma \gamma$ signal and the continuum $gg\to
\gamma\gamma$ QCD background, which can alter the production rate as well as
modify the line-shape and apparent mass. We find a modest enhancement by $\sim
20$% under favorable circumstances, for a large $\Phi$ width. The effect of
interference on the apparent scalar-pseudoscalar mass difference in a
two-Higgs-doublet model is found to be also modest. An exploratory study
indicates that similar effects are to be expected in the $gg \to \Phi \to Z
\gamma$ channel. In this and other models with a large $\Phi$ total width, the
dominant $\Phi$ decays are expected to be into $t \bar t$ final states. We
therefore also study the effects of interference of the $gg\to \Phi \to t\bar
t$ signal with the $gg\to t \bar t$ continuum QCD background and show that in
the presence of standard fermions only in the $gg\to \Phi$ loops, it is
destructive causing a dip in the $t \bar t$ mass distribution. Including
additional vector-like quarks leads to a different picture as peaks followed by
dips can then occur. We use the absence of such effects in ATLAS and CMS data
to constrain models of the production and decays of the $\Phi$ state(s).
