A search for heavy pseudoscalar ($A$) and scalar ($H$) Higgs bosons decaying into a top quark pair ($t\bar{t}$) has been performed with 20.3~fb$^{-1}$ of proton--proton collision data collected by the ATLAS experiment at the Large Hadron Collider at a center-of-mass energy of $\sqrt{s} = 8$ TeV. Interference effects between the signal process and Standard Model $t\bar{t}$ production, which are expected to distort the signal shape from a single peak to a peak--dip structure, are taken into account. No significant deviation from the Standard Model prediction is observed in the $t\bar{t}$ invariant mass spectrum in final states with an electron or muon, large missing transverse momentum, and at least four jets. The results are interpreted within the context of a type-II two-Higgs-doublet model. Exclusion limits on the signal strength are derived as a function of the mass $m_{A/H}$ and the ratio of the vacuum expectation values of the two Higgs fields, $\tan\beta$, for $m_{A/H} > 500$ GeV.