  We discuss the production of a heavy neutral Higgs boson of a CP-conserving
Two-Higgs-Doublet Model in gluon fusion and its decay into a four-fermion final
state, $gg (\rightarrow VV) \rightarrow e^+e^-\mu^+\mu^-/e^+e^-\nu_l\bar\nu_l$.
We investigate the interference contributions to invariant mass distributions
of the four-fermion final state and other relevant kinematical observables. The
relative importance of the different contributions is quantified for the
process in the on-shell approximation, $gg\rightarrow ZZ$. We show that
interferences of the heavy Higgs with the light Higgs boson and background
contributions are essential for a correct description of the differential cross
section. Even though they contribute below $\mathcal{O}(10\%)$ to those heavy
Higgs signal cross sections, to which the experiments at the Large Hadron
Collider were sensitive in its first run, we find that they are sizeable in
certain regions of the parameter space that are relevant for future heavy Higgs
boson searches. In fact, the interference contributions can significantly
enhance the experimental sensitivity to the heavy Higgs boson.
