Searches are presented for heavy scalar ($\mathrm{H}$) and pseudoscalar ($\mathrm{A}$) Higgs bosons posited in the two doublet model (2HDM) extensions of the standard model (SM). These searches are based on a data sample of $\mathrm{p}\mathrm{p}$ collisions collected with the CMS experiment at the LHC at a center-of-mass energy of $\sqrt{s} = 8~\mathrm{TeV}$ and corresponding to an integrated luminosity of $19.5~\mathrm{fb}^{-1}$. The decays $\mathrm{H} \rightarrow \mathrm{h} \mathrm{h}$ and $\mathrm{A} \rightarrow \mathrm{Z} \mathrm{h}$, where $\mathrm{h}$ denotes an SM-like Higgs boson, lead to events with three or more isolated charged leptons or with a photon pair accompanied by one or more isolated leptons. The search results are presented in terms of the $\mathrm{H}$ and $\mathrm{A}$ production cross sections times branching fractions and are further interpreted in terms of 2HDM parameters. We place 95% CL cross section upper limits of approximately $7~\mathrm{pb}$ on $\sigma \mathcal{B}$ for $\mathrm{H} \rightarrow \mathrm{h} \mathrm{h}$ and $2~\mathrm{pb}$ for $\mathrm{A} \rightarrow \mathrm{Z} \mathrm{h}$. Also presented are the results of a search for the rare decay of the top quark that results in a charm quark and an SM Higgs boson, $\mathrm{t} \rightarrow \mathrm{c} \mathrm{h}$, the existence of which would indicate a nonzero flavor-changing Yukawa coupling of the top quark to the Higgs boson. We place a 95% CL upper limit of 0.56% on $\mathcal{B}(\mathrm{t} \rightarrow \mathrm{c} \mathrm{h})$.