We combine static magnetic susceptibility $\ensuremath{\chi}(T)$, muon-spin relaxation, and $^{1}\mathrm{H}$ nuclear magnetic resonance measurements to explore the spin dynamics in the disordered-induced quantum spin liquid candidate ${\mathrm{H}}_{3}{\mathrm{LiIr}}_{2}{\mathrm{O}}_{6}$. Inverse Laplace transform analysis of the $^{1}\mathrm{H}$ spin-lattice relaxation rate $1/{T}_{1}$ enables us to identify two characteristic temperatures ${T}_{\mathrm{g}}=110\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and ${T}^{*}=26\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. Below ${T}_{\mathrm{g}}$, a slower $1/{T}_{1}^{\mathrm{slow}}$ component dictated by gapped excitations with a spin gap ${\mathrm{\ensuremath{\Delta}}}_{h}=30--38\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ evolves distinctly from a faster $1/{T}_{1}^{\mathrm{fast}}$ component pertaining to gapless excitations. Furthermore, we observe a sub-Curie divergent $\ensuremath{\chi}(T)\ensuremath{\propto}{T}^{\ensuremath{-}0.68}$, a power-law dependent $1/{T}_{1}^{\mathrm{fast}}\ensuremath{\propto}{T}^{1.4}$, and a weakly activated $1/{T}_{1}^{\mathrm{slow}}\ensuremath{\propto}exp(\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{l}/{k}_{\mathrm{B}}T)$ (${\mathrm{\ensuremath{\Delta}}}_{l}=3--6\phantom{\rule{0.16em}{0ex}}\mathrm{K}$) below ${T}^{*}=26\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. All these features suggest the coexistence of a disordered spin-liquid state and spin singlets with spatially distributed gaps.